DUKE MATHEMATICAL JOURNAL

EDITED BY

ARTHUR BYRON COBLE DAVID VERNON WIDDER JOSEPH MILLER THOMAS Managing Editor

WITH THE COOPERATION OF

H, E, BRAY L. R. FORD R. E. LANGER J. A. SHOHAT L. W. COHEN J. J. GERGEN C. C. MacDUFFEE G. T. WHYBURN E. P. LANE OYSTEIN ORE

AND THE MATHEMATICS DEPARTMENT OF DUKE UNIVERSITY

VOLUME 1

1935

DUKE UNIVERSITY PRESS DURHAM, N., C.

CONTENTS

Voice 1, 1935

Apams, C. R. and Lewy, Hans. On convergence in length............ 19 BEcKENBACH, E. F. On subharmonic functions........................ 480 BrrkuorrF, GARRETT. Orthogonality in linear metric spaces............. 169 BirkuHorr, G. D. and Hesrenes, M. R. Natural isoperimetric condi- TT ae 198 Generalized minimax principle in the Calculus of Variations. .......... 413 Boas, R. P., Jr. Necessary and sufficient conditions in the moment prob- a oe hla rad i ich aany Sn alia regalia 449 BraHana, H.R. Metabelian groups and trilinear forms................ 185 BucuanaN, H. E. and Duren, W. L., Jr. On the characteristic exponents in certain types of problems of mechanics........................5. 436 Cameron, Rosert H. Linear differential equations with almost periodic hn cs nabs Sehd Caasina a oRAn seeds heb eeb cme ebels 356 Caruitz, LeEonaRD. On certain functions connected with polynomials in Oe er Pee ee re a ee ee ee 137 On the representation of a polynomial in a Galois field as the sum of an i iia cece we Chek eins aah ae ce ae dws 298 Dickson, L. E. Linear algebras with associativity not assumed......... 113 Dorwart, H, L. Concerning certain reducible polynomials............. 70 Downs, Tuomas L., Jr. Asymptotic and principal directions at a planar i icine eked $50 GS SADEEE UR TEES RARE ENS Smee 316 Duren, W. L., JR. and Bucnanan, H. E. On the characteristic exponents in certain types of problems of mechanics.......................-. 436 Frvan, Epwarp J. On the number theory of certain non-maximal domains of the total matric algebra of order 4...............00000 cece cues 484 Forp, L. R. On properties of regions which persist in the subregions bounded by level curves of the Green’s function.................... 103 Frame, J. 8. Some irreducible monomial representations of hyperorthog- in cicidons vannbn dunn nhane Mebea maces Ghanian ses eens 442 Hestenes, M. R. and Brrkuorr, G. D. Natural isoperimetric conditions ee Ge I . ciiccuncebecnbenebaeshtareeancaanes 198 Generalized minimax principle in the Calculus of Variations........... 413 Humpureys, M. Gwenetu. On the Waring problem with polynomial NG a tg pinid 6'> cin Ske eb eee ae na ee ene ae ak ae maine nee eee 361 vAN Kampen, Eapertus R. On some characterizations of 2-dimensional cS icons atedne eennedkevad 65s okeh sib deaAeeebin sok ebed 74 iii

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iv CONTENTS

LaNE, Ernest P. The neighborhood of a sextactic point on a plane curve. . 287 LATIMER, CLAIBORNE G. On the fundamental number of a rational gener-

EE eS a er ee ee 433 LerscHETz, 8. Chain-deformations in topology........................ ‘1 ei ete es cig din dé Cw Oah ad a waa ee OES 392 LEvINE, J. New identities in conformal geometry...................... 173 Lewy, Hans and Apams, C. R. On convergence in length.............. 19 Lowan, Arnotp N. Heat conduction in a semi-infinite solid of two differ- Is rae ous OVER A TS bt Os a yeas Mae we tere daeabnee sa a's 94 MacQueen, M. L. On certain properties of projective parallelism of tela nek Oe Reha A Sede} Ge eek ColaE eed eek as Nees 496 Marta, Aurrep J. Concerning the equilibrium point of Green’s function eden coeur bees ou iiy peated te oie tate ei as bad os 491 Moore, Cuartes N. On criteria for Fourier constants of L-integrable en a i ae cen keene kek bes eee’ 293 | Myers, SumNer Byron. Riemannian manifolds in the large............ 39 Connections between differential geometry and topology. I. Simply RE RE Oe BE ERE Ee oe roe ee RC ey Ew 376 NatHaNn, D. 8. One-parameter groups of transformations in abstract Ps raven dc Cue dkgide SNA eVeues sels Cevedesdeenh habeus 518 RanvE.ts, W.C. On Volterra-Stieltjes integral equations............... 538 RicHarpson, Moses. On the homology characters of symmetric products. 50 Rosser, J. B. A mathematical logic without variables IT.............. 328 Sietey, D.T. Groups involving five complete sets of non-invariant conju- gate operators........ Sika a Rees ERA eeada Cera se nner seawae 477 Synce, J. L. On the neighborhood of a geodesic in Riemannian space.... 527 SzAsz, Orro. Generalization of two theorems of Hardy and Littlewood ET OSLO OT ET TE EE TT PE SE EET TET EET 105 Terry, Henrietta. Abelian subgroups of order p” of the J-groups of the abelian groups of order p" and type 1, 1, --- ..............0000. 27 Wuirney, Hasster. A function not constant on a connected set of i tee Ae hea kad ae aee eae 46 aN SOOT 514 Wuysurn, G. T. Generalized perfect sets...................2-0000005 35 Wipper, D. V. An application of Laguerre polynomials................ 126

Wiper, R. L. On locally connected spaces..................00 2c eeee 543

Be

re

CHAIN-DEFORMATIONS IN TOPOLOGY By S. LEerscHetz

In topology one has repeated occasion to consider homotopic deformations of chains. They give rise to a basic boundary relation' between the extreme positions of a chain c, in the homotopy and what might be termed the loci of c, and of its boundary F(c,). All the consequences of the homotopy that concern algebraic topology (i.e., boundary relations and the associated homologies) may be derived from the fundamental relation. It seems natural therefore to call chain-deformation any scheme wherein two p-chains c,, c, and two other chains that are to take the part of the loci mentioned above, satisfy a boundary relation formally identical with the fundamental relation of homotopy.? This notion has already been exploited in a recent paper.* We return to it here, first to develop it more fully and then to apply it to the study of the sets that are obtained whenever, in the definition of locally connected sets, singular cells and spheres are replaced by chains. These new sets may be described as locally connected in the sense of homology, and their types correspond substantially to the locally connected types that we have recently investigated. The pas- sage from the first class to the second corresponds also to a substitution of chain-deformation for homotopy.

One of the important results of L2 was the identification of certain locally

Received February 12, 1935.

1 Given for the first time in our Colloquium Lectures, Topology, New York, 1930, p. 78.

2 While chain-deformations have most of the properties that their name suggests, they are essentially different from homotopy. This is clearly seen by noting the different effect in the very simple case of the circuits on an orientable surface of genus p = 2. Homotopy leads, in this case, to the non-commutative Poincaré group, chain-deformation to the much simpler abelian group with 2p free generators.

3S. Lefschetz, On generalized manifolds (= L1 in the sequel), American Journal of Math- ematics, vol. 55 (1933), pp. 475-499.

4S. Lefschetz, On locally connected and related sets (= L2 in the sequel), Annals of Mathematics, vol. 35 (1934), pp. 118-139. We call attention to the following errata: p. 119, line 23, replace LC by LC®; p. 126, suppress line 3 from bottom; in line 4 from bottom, sup- press “‘convex’’; in line 5 from bottom, replace ‘“‘convex sets of §’’ by “‘spheres’’; p.127, line 13, replace K* by &*.

Local connectedness in the sense of homology was introduced by P. 8. Alexandroff in his paper: Untersuchungen tiber Gestalt und Lage abgeschlossener Mengen beliebiger Dimension, Annals of Mathematics, vol. 30 (1929), pp. 101-187. See also his recent paper: On local properties of closed sets, Annals of Mathematics, vol. 36 (1935), pp. 1-35, §3. The same prop- erty for euclidean domains plays a central part in R. L. Wilder’s recent work. See in par- ticular his last paper: Generalized closed manifolds in n-space, Annals of Mathematics, vol. 35 (1934), pp. 876-903.

1

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2 S. LEFSCHETZ

connected sets with the absolute neighborhood retracts or absolute retracts in the sense of Borsuk. (See in this connection footnote 13.) A similar identifiea- tion is possible here with generalized retracts in the sense of chain-deformations. Rather than to press this analogy, we preferred to investigate the mutual rela- tions between the two kinds of local connectedness as well as with Borsuk’s locally contractible sets, but there remains still much to be done along that line.

One of the most useful notions introduced in L2 was that of the semi-singular complex (singular complex with only part of the expected cells present). It is extended here to aggregates of chains related like the oriented cells of a complex and is found no less useful in the present investigation.

In the endeavor to free our results, as far as possible, from any specific choice of chains, we have presented the theory of chains in axiomatic form at the beginning of the paper. This has the additional advantage of making the paper less dependent upon our previous writings.

Our general notations are those of Topology, with the abbreviations of L1 and L2: LC, and NR stand for “locally connected”’ and “neighborhood retract.”’ In addition we shall write HLC for ‘‘LC in the sense of homology’’, and similarly for HNR. Our two other abbreviations are c.s.v.t. for “continuous single- valued transformation’’, f.c.o.s. for ‘finite covering by open sets’. The reader will have no difficulty in getting accustomed to these alphabetical notations whose advantages, after all, need not be reserved for the political domain.

§1. The chains of a topological space

1. There are various ways of extending to a topological space ® the basic properties of the chains of a geometric complex, their cycles, their boundary relations and the like. Regardless of the procedure adopted certain properties are preserved. As it is with these common properties that we are chiefly con- cerned, we shall recall them briefly and state them as postulates for the chains of KR:

I. There exists for every p = 0, 1, --- , a set of topological invariants of KR, its p-chains c,, and their collection {c,} constitutes a free additive abelian group.

II. There exists an operation F defined topologically for all the chains and such that F {c,} is a homomorphism of {c,} into {cp-1}, and of {co} into the identity.

F is the boundary-operator, Fc, or F(cp) is the boundary of cp, their mutual relation being indicated by a boundary relation,

(1.1) Cp — F(cp),

5 These properties are fully developed in Topology, Chapters I, II. For a more sys- tematic exploitation of the abstract viewpoint see A. W. Tucker, An abstract approach to manifolds, Annals of Mathematics, vol. 34 (1933), pp. 191-243, where further references, notably to the papers of W. Mayer, will be found. It is to be noted that whereas they con- sider only abstract complexes and chains, we have always tied them up with definite point- sets. It might be advisable to use different terms, such as abstract complex or chain, geometric complex or chain, for the two concepts.

CHAIN-DEFORMATIONS IN TOPOLOGY 3

while to express merely that c, is a boundary we write a homology, (1.2) Cp ~ 0.

The chains c, such that Fc, = 6 are called p-cycles, and generically denoted by yp. In particular, every cois ayo. From II follows that {y,} is a subgroup of {cp}.

Ill. FF = 0.

This means that boundaries are cycles. Moreover if 8, is a generic boundary, from II follows again that {8,} is a subgroup of {yp}. The difference-group (factor-group of the customary terminology) {yp} — {8} is the pt® homology group of R. When it is a free group the number F,,(®) of its independent generators is called the p** Betti-number of ®.

It is to be kept in mind that in the present paper, all homologies imply bounding. This is the reason why we use the symbol ~ and not =, for example as in Topology, Chapter VII, or Ll, for yp = 0 merely meant that y, was a finite or infinite sum of bounding cycles, or neglected chains, without being itself strictly in one or the other category. From the group viewpoint, our earlier procedure corresponds to topologizing the groups, replacing {8p} by its closure, say {8,}, and taking as p homology group the difference-group lyp} = {Bp}.

IV. There exists a numerical topological invariant linear function of zero-chains, the Kronecker-index (co), and (co) = 0 when co ~ 0.

In the case of complexes (co) is the number of points of co each counted with its coefficient in the expression of the chain.

V. With every c, there is associated a unique closed subset | c, | of R such that:

(a) [O,}=0; (db) lep+e,|Cle|+le|, (c) | F(cy) | Cl ep]; (d) dim | c, | 2 p when c, ¥ 0.

If A is any closed subset of R we say that c, C A whenever |c,| CA. Taken together with V, this enables us to define the boundary relations, homologies and homology-groups mod A, or relative relations, by contrast with the previous type called absolute.

From V(d) follows that when dim R = n is finite every cp, = 0 for p > n. In particular there are no homology groups, absolute or relative, for dimensions > n.

2. There remains one more property, but it is most conveniently expressed in terms of the very useful notion of quasi-compler. A quasi-complex & is a collection of chains of ® such that: (a) its p-chains form a subgroup of {c,}; (b) F R CR: when c, C KR likewise Fc, C R. The quasi-complex is finite whenever the dimension of its chains is bounded, and in addition for every p there is a finite base for its p-chains whose elements are independent. In that case we frequently reserve the name “‘chain of R”’ for the chains of the bases, and call subchains of R the other chains of the quasi-complex.

If c; is any base chain of a finite &, the chains c}_, entering in the composition of F(c}), those entering in the composition of F(c}_,), ete., are called the boundary-chains of c}. The sum | ci} | + 2|c3_,| + --- has a least upper bound called the mesh of &.

4 S. LEFSCHETZ

Our last axiom may now be stated:

VI. (Subdivision axiom.) Every chain of R is a subchain of a finite quasi- complex R whose mesh is arbitrarily small.

The complex & is called an elementary decomposition of the chain, an ele- mentary «-decomposition when its mesh < e.

3. The following are noteworthy examples of systems of chains for which all the axioms hold:

(a) ® is a topological space and its chains are the singular chains on ® in the sense of Topology, Chapter II,® that is to say, the linear forms in the singular cells on ® with coefficients members of an additive abelian group 2. All the axioms except the last are readily verified, and the last is verified also provided that we agree to identify a singular c, with all its subdivisions and call chain the class thus obtained. Otherwise we merely have as a theorem that all the members of the class are equivalent regarding boundary relations and homol- ogies (Topology, p. 88). It is understood of course that each cell of a subdivi- sion of c, is to be oriented concordantly with the carrying cell of c,.

A noteworthy case is that where ® is a simplicial complex K. It is then shown that the homology groups derived from the chains made up of the oriented simplexes of K, or combinatorial homology groups, are isomorphic with those defined above and hence the former are topological invariants.

(8) Mis a compact metric space and the chains are the projection-chains of L1. They are certain specific subchains, taken here with coefficients in a field M, of a fundamental infinite complex K which consists of the skeleta #* of f.c.o.s. together with their joining cells (L1, p. 470-472).”. The configuration K = ® + K may be identified for convenience with its topological image on the Hilbert parallelotope § (see loc. cit. for details). The projection-chains C,,, of K determine the chains c, of ®, with |c,| = |Cpii| - R(L1, p. 477). Forp = 0 we define the Kronecker-index by (co) = (F(C.)) which is equivalent to the definition of L1, p. 489.

We shall call chains of type (a) singular and chains of type (8) regular. Unless otherwise specified regular chains shall be the type usually considered in the sequel.

The only non-compact metric spaces that interest us are the separable locally compact spaces. For these we may consider the so-called finite cycles, or cycles on self-compact subsets of the space, and they are the only kind needed later.

Our definition of regular chains is not intrinsic in that it depends upon a specific funda- mental complex K. It may be made intrinsic as follows. Suppose that we pass to a new fundamental complex K’. By means of the deformation theorem of Topology, p. 328 we

6 See also S. Lefschetz, On singular chains and cycles, Bulletin American Mathematical Society, vol. 39 (1933), pp. 124-129.

7 In the construction of K (loc. cit., beginning of No. 2) the finiteness of dim ® plays nordle. All that matters is that the number n is the least order of any « —f.c.o.s.

CHAIN-DEFORMATIONS IN TOPOLOGY 5

first reduce a projection-chain C,,, of K defining c, of R to a chain C},,, of K’, then by Theorems IV, V of Li (p. 484), we reduce C;,; to a projection-chain (%,, of K’. The construction, which depends upon a choice of bases as in L1, No. 16, is made unique by adopting a fixed numbering of all the cells of K’. It is the class of all projection-chains C”,,; thus obtained which is c, under the new definition. It is not clear, however, that | c, | is independent of K. Let us assume that this is not necessarily so and let A, A’ be the sets as determined respectively by K and K’. We have immediately A C A’. If we return from K’' back to K we show readily that we reduce C}+, to a projection-chain on a fundamental complex for A necessarily identical with C,,,. Hence A’ C A and therefore A’ = A.

It may be observed that projection-chains are imposed by the subdivision axiom. They are in fact the only chains of K which we are able to subdivide into elements of the same nature, notably as regards the subdivision of the elements C, into elements with the proper C, boundaries. On the other hand their uniqueness is only proved when QM is a field, so that the restriction imposed upon IM is largely due, in the last analysis, to the subdivision axiom.

Besides the two classes (a), (8) others may well be introduced. We mention notably: (y) the Vietoris-chains for compact metric spaces;? (6) the chains recently defined by Cech? for spaces even more general than topological spaces. In the case (y) there is an associated group I as general as in (a), while in (6) M is again restricted to a field. In fact, when M is a field the homology theories yielded by (8), (vy), (6) are the same. Furthermore when ¥ is a field, and ® is LC” in the sense of L2 the four types coalesce (Topology, p. 333).

All but the last of our axiomatic properties are readily verified for the four types mentioned. Regarding the last, subdivision, for the singular type it is a consequence of the definition and for the regular type it is proved as in LI, p. 490. Since they are the only two considered in the present paper this will suffice for our purpose.

§2. Chain-deformations

4. As a matter of convenience we shall assume henceforth that the basic space ® is separable, metric, locally compact, and later even, a good part of the time, that it is compact. Until further notice the chains are finite chains in the sense of No. 3 and, unless otherwise stated, merely subjected to our basic axioms.

In the applications, even when dealing with a single chain, one is constrained to deform a whole quasi-complex. For example to have the analogue of e- homotopy, we must break up the chains into the elements of a & of mesh e which is then to undergo the deformation. Owing to this it is best to define at the outset a chain-deformation & of a quasi-complex R = {c,}, into another R’ = {c,}. We understand then by such a chain-deformation, (merely “def- ormation”’ when there is no ambiguity), a homomorphism of the chain-groups of R into those of the same dimension for &’, which commutes with F and which is associated with a linear operator D on the chains of R having the property that

8 L. Vietoris, Uber den hiheren Zusammenhang kompakter Réume und eine Klasse von zusammenhangstreuen Abbildungen, Math. Annalen, vol. 97 (1927), pp. 454-472.

°E. Cech, Théorie générale de UV homologie dans les espaces abstraits, Fundamenta Math- ematicae, vol. 19 (1932), pp. 149-183.

6 S. LEFSCHETZ

whatever c, of R, De, is a uniquely defined (q¢ + 1)-chain of R, called the defor- mation-chain of c,, such that

(4.1) FDe, = 0ce,—¢,— DF cy. Written as relations between operators on R we have then (4.2) oF = F3#,

(4.3) FOD+DF=08-1.

D may also be considered as inducing for each g a homomorphism of the group of the qg-chains of R into the group of the (¢ + 1)-chains of ® satisfying (4.3). The preceding properties, and in particular (4.1), written also

(4.4) Deg ve, —¢, — DFe,),

are those satisfied by chains under a homotopy (Topology, p. 78) which we are thus merely taking as postulates for chain-deformation.

5. A cycle y,, a chain c, + its boundary F(c,), form special quasi-complexes. For the former 3 merely demands that

(5.1) FD vp = Pyp — Yp-

Since a boundary is a cycle, dy, — vy, is a cycle, and hence #y, is one also. Therefore the chain-deform of a cycle is a cycle. Regarding c,, 8 demands two deformation-chains D c,, D Fe, such that

(5.2) F De, = dep — cy — D Fe, (5.3) FD Fe, = Foc, — Fep.

Of these relations the second follows from the first, since it merely expresses the fact that the boundary of the right-hand side in (5.2) is a cycle. Therefore for c, also a chain-deformation is specified by the single relation (5.2). It is clear that in the sense just considered a chain-deformation of a quasi-complex & induces a chain-deformation of its individual chains.

6. Whenever &, &’ and the D-chains are on a given set A we say that the chain- deformation is over A. The chains in any given set B form a quasi-complex, and its chain-deformations are called chain-deformations of B. Thus if 3 deforms all the chains of B into chains of a set B over A, we say that B is chain-deformed into B’.

Whenever &, &’ are of mesh < ¢ and all the chains D involved are of diameter < ¢, we say that # is an ¢ chain-deformation of & or of any subchain of R. An e chain-deformation of c, consists in imposing upon it an elementary e-decomposi- tion reducing it to a R of mesh < e, and then ¢ chain-deforming &.

CHAIN-DEFORMATIONS IN TOPOLOGY 7

7. Let # be as before and let 3’ deform &’ into R’’ with D’c, as the deforma- tion-chains. We have now a transformation 3’’ = 8’8 of R into R”’ and we shall show that it is a chain-deformation. Clearly #’’ is a homomorphism com- muting with F so (4.3) alone must be verified. We have

(7.1) DF +FD'=0'-1

operating on &’, but considered as operating on & this must be written (7.2) D’Fd + FD'S = 0'8 — 8 = 8" — BV.

Adding (7.2) and (4.3), we have

(7.3) (D + D’d)F + F(D + D’d) = 8” —- 1.

Hence if we introduce the linear chain-operator D’’ = D + Dd, we have (7.4) D"F + FD" = 0” —1,

so that (4.3) holds, with D’’ as the D operation. From its definition D’’ is linear, hence D”’ is a deformation-chain.

Clearly # = 1 is a chain-deformation corresponding to D = 0. Regarding the inverse 3 it can only be defined, if at all, when # is one-one. In particular, this holds when & consists of a single chain plus its boundary. When #@ is one- one c, = vc, determines uniquely c, and hence Dc,. If we set

i , De, = Dc, = -—De,

we have D'dFc, = D’Fdc, = —DFc,, and therefore

FD'c, =C¢,—C, — D F(c,).

This shows that the passage from &’ to & is a chain-deformation with —Dc, as its deformation-chain. Thus in the case in question 3~' is a chain-deformation.

If we call two q-chains of R equivalent whenever one of them can be chain- deformed into the other, the results of the two preceding paragraphs show that this equivalence is transitive, reflexive, and symmetric.

8. Homotopy and chain-deformation. Let a set A undergo a homotopy T into a set A’ over ¥ and let B be its locus throughout the deformation. If c, is a chain of A the homotopy will determine a chain ¢, = Tc,. One suspects intuitively that, as is obvious for singular chains, c, is a chain-deform of c, over B. We shall in fact prove:

THEOREM I. When the chains adopted are regular or singular a homotopic deformation of a compact™ set induces a chain-deformation of all its chains.

10 We mean here that the set is compact as a space, or, as it is sometimes called, self- compact.

8 S. LEFSCHETZ

For singular chains the theorem is practically proved in Topology, p. 79, so that we may limit our treatment to regular chains. This being the case, we may clearly replace A, B, A’ by | ¢, | , its locus, and 7’- | c, | , and hence assume A, A’ closed and the locus the whole space, that is B = 8. Under the cir- cumstances let L be a fundamental complex for A which is a subcomplex of K (L1, p. 474) and let the whole configuration be assumed immersed in the Hilbert parallelotope 5 (No.3). Let P', P?, --- be the vertices of K, which we take to be points of A and consider the product L X \ = L*, where ) is a unit- segment. It is an infinite convex complex, which we subdivide in such a manner that if Z is any cell of L, the cell E X \ of L* becomes a sum of cells E X ’, where )’ is an interval of \. That is, every ‘“‘prismatic” cell of L* is subdivided by sections “parallel’’ to its bases. This is carried out in such manner that if the cells of K, numbered in some order, are E', E?, ..- , then E* X } is sub- divided into cells equally spaced (i.e. with intervals \’ of equal length) and whose number > 2°. Finally we subdivide in any manner L* simplicially without introducing new vertices, which may always be done since its cells are all convex. We continue to call L* the ultimate complex. On L* we have a representative Cy+: of c,, and also a chain C ol ,, obtained by translation from C,,; and deforma- tion-chains D C,4:, DFC,+: such that

96,4 ~C... — Cau = DPC».

We shall now transform L* barycentrically as follows. Let us suppose that the deformation T of A into A’ depends upon a parameter ¢ varying from 0 to 1. On the cell P* X \ there will be a certain number of vertices P®, .-- , P*" of L*. Mark now on the path of P* as ¢t varies, the positions of P* corresponding to the values z/r, (¢ = 0, 1, --- , 7), and let them be P* = Q®, --- , Q*". We map P*‘ into Q*‘, thus obtaining a unique image for every vertex of L*, and for every simplex with certain vertices P*' we insert the corresponding simplex with vertices Q*'. The various chains on L* are thus mapped into chains defining chains, of one dimension less on §%, in the same sense as those on K (Topology, p. 324). Furthermore by a deformation in §, (Topology, p. 332) together with Theorems IV, V of L1, all leaving C,,; invariant, we may reduce them to projection-chains of K. By the very definition of the different chains, D C,+1 and Cow go into chains of K which define a D c, and c, satisfying (4.1), with |De,| CR, |c, | CA’. This proves the theorem.

§3. Retraction properties

9. Henceforth we restrict the chains definitely to the regular or singular types, and until further notice, we also assume that the space 9 is compact, metric.

We say that a set A is chain-shrinkable onto a subset B whenever every one of its chains is deformable onto B over A. We also say that B is an HR of A. The HR property is the analogue of the retract property, however, with the difference that in retraction of A onto B under a c.s.v.t. the set B remains fixed point for point, while no such condition is imposed under chain-shrinking. The difference is more apparent than real as shown by:

CHAIN-DEFORMATIONS IN TOPOLOGY 9

TuHeoreEM II. Jf c, is chain-deformable onto a closed set A on R, the chain- deformation 8 of c, onto A may be so chosen as to be merely over R — A. More precisely it may be associated with an e-elementary decomposition, € assigned, {c;} of cp, such that the elements c?‘ alone meet R — A and that the others are not deformed: dc3*** = c2*** (8 = Lon A).

The basic step is the derivation of the elementary «-decomposition. When the chains are singular we merely take a subdivision of mesh ¢ and call c3‘ its elements on A and c?*** the rest. The problem is more difficult when the chains are regular.

Let first L be a fundamental complex of A which is also a subcomplex of K (L1, p. 474) and let there be given a chain-deformation # of c, associated with an

elementary e-decomposition &’ = {c,‘} and the deformation-chains D’ c;. If C; +, is the representative of c,' in K and c; ¢ A, we may write (9.1) on = Cli + Coit,

; 4g ; ° es ° where C7}, = L-C,}, and C77" is the remaining part. The two chains at

the right of (9.1) determine chains c?‘, c}*** such that

(9.2) e,' = ff + a, where c;‘*' alone meets R — A. If cy CA, we set cf! = c, cg = 0. In any case c3', c?** are both on | c,‘ |, hence their diameters < ¢, so that R = {c,}

is likewise an « elementary decomposition of c,. Moreover except for the exist- ence of a suitable # it behaves as demanded by the theorem. We shall now construct #.

Regarding the operation D to be associated with # we first specify that De?! = 0. Next we treat D’c?* like c,‘ above and obtain

(9.3) D’ . = dj 41 + ds +1)

where di,, C A, di, CR — A, and we now set Dc? = dij,. We now de- termine the dhain-decmnpectiien fc, | of the new Giathons chain dc, by the boundary relations

(9.4) De, > dc) — ci — DFC),

in which for every combination 7, g, the only unknown term is c, * = dc}, which is thus uniquely determined by (9.4).

Since the chains F(c*‘) are of c*‘ type, their D in (9.4) are both zero, hence c,?* = ¢?', Similarly we have c,?‘*! CR — A. Finally,

a

(9.5) D' ci +8’ ci —ci —D' Fe} for every value of 7. Hence

(9.6) (D' — D) cf +0’ — 9) ci — (D’ — D) Fe!

10 S. LEFSCHETZ

Since the chains (D’ — D) c} C A, 8’ c} C A, (9.6) shows that dc} C A also and hence 3c, C A. Therefore all the conditions of the theorem are effectively fulfilled.

Remark. Since | Dc} | C|D’c; |, if the initial chain-deformation 0’ is ¢, so is the modified one #.

Corotiary. If A is open instead of closed, we may choose {ci} such that e! CA,c2**! CR-A,withdc?' =c?' (8 = 1onA).

For all that is necessary is to apply the theorem to A.

AppuiicaTion. If F(c,) C A the chain-deformation may be so chosen as to leave F(c,) unchanged.

10. The notion of chain-shrinking as here presented suffers from the dis- advantage of not being local. The “local” properties are, however, frequently the most important, and so we shall consider them now.

Let A, B be subsets of R. We say that A may be chain-shrunk away from B whenever there is an open set U D B such that A may be chain-shrunk onto A—U. We have at once

THEOREM III. Jf A may be chain-shrunk away from every point of a compact set B, it may be shrunk away from B.

By the Borel covering theorem, B has a f.c.o.s. {U*}, such that A is shrinkable onto every A — U‘. We can find another f.c.o.s. {V‘} of B such that Vi CU‘ We shall use both coverings simultaneously.

By assumption A may be chain-shrunk onto B — U' and as a consequence every chain c, of A will have been displaced to the exterior of a certain open set Wi'D> V'.B = B.

Suppose that we have succeeded in showing that the displacement may be carried out similarly to outside a certain open set

Wed Be = (V1 4+... + ‘VE)-B.

If W*' D> Bweare through. In the contrary case, there is at least one more V, which we may assume to be V* such that V*.B € W*". In any case however the chains of A not on W*~' will be at a positive distance 5 from B*'. By assumption we can displace all the chains of A onto A — U*. However by Theorem II, this displacement may be replaced by one onto the exterior of the spherical neighborhood S(U*-B, 1/26). The chains thus displaced will be at a positive distance from B*, hence outside of some W* D> B*. Proceeding thus we shall ultimately have a W" > B, with all chains deformable onto A — W". The theorem is therefore proved.

§4. HLC spaces and their relations to other spaces

11. The definition of the HLC sets of various types is entirely similar to that of LC sets. Here also, however, we need the local characterization.

11 Menger, Dimensionstheorie, p. 160.

CHAIN-DEFORMATIONS IN TOPOLOGY il

We shall say then that § is:

p-HLC at the point x whenever every open set U > 2 contains another V > zg also, such that every p-cycle, p > 0, of V is ~ 0 on U, and every zero- cycle yo of V is ~ on U toa point of V taken (yo) times;

HLC? at x whenever it is g-HLC for every ¢ S p;

HLC% at x whenever it is g-HLC for every g whatsoever;

weak HLC at x whenever V can be determined as above independently of gq.

The strong HLC (merely HLC) will be defined for a compact metric ® in No. 15.

Finally ® itself is p-HLC, --- whenever it has the corresponding property at all its points.

The ¢, 7 formulations in the compact metric case are as usual: in place of “for every U there is a V” we must have “for every « > 0 there is an 7 > 0”. In particular the HLC? condition may be reduced to “for every ¢ there is an n(e, p) such that every q-cycle gq < p (a yo whose (yo) = 0) of diameter < 7 bounds a chain of diameter < e’’, while the weak HLC condition will be of the same form with 7 independent of p.

12. If c, and c}_, are two chains of a finite quasi-complex &, we shall say that they are incident whenever c}_, is a boundary chain of c}. The aggregate of the specifications of the incidences of & is called the patiern of &.

Suppose that we have given a R whose pattern we are to reproduce on t, and of all the elements expected let there be present all the zero-chains and some, but not necessarily all, of the rest, so that the chains present form a quasi-sub- complex R’ of R. We call &’ a partial realization of R. Let us suppose that out of an expected c} and its boundary chains, there are already present the chains c// in 8’. We call max diam = | c,/ | the mesh of the partial realization R’ of K.

If cé is an expected chain of and F(c}) is already present in &’ a necessary condition is that it bea cycle. Furthermore if q = 1 and if its imposed boundary relations are

(12.1) Ci > 5 C0 where, by assumption, the chains c) are already present, from (12.1) follows (12.2) nj (ch) = 0,

and these relations must be satisfied if the data are consistent. They shall naturally be assumed if &’ is given, or must be verified whenever a &’ is con- structed.

13. THrorem IV. N.a.s.c. for a compact metric space R to be HLC? are that for every « > 0 there exist an ne, p) > 0 such that every partial realization R' on MR of a Ry +1 whose mesh < n(e, p) may be completed to make up the expected R, +1 of mesh ¢.'*

12 Compare with the analogous proposition for LC?-sets, L2, p. 120.

12 S. LEFSCHETZ

Since F(c,), g S p, is a special R’ corresponding to a R which consists of c, and F(c,), the condition of the theorem is clearly sufficient for an HLC?. We must therefore merely show its necessity, and as it is trivial for p = —1, we may take p > 0 and use induction on p.

Under our assumptions the theorem holds for p — 1, and there is a corre- sponding n(e, p — 1). Moreover since 8 is p-HLC we have also the constant £(e, p) of the p-HLC property (n(e, p) in the definition).

Let the mesh a@ of R’ be < (8, p — 1) < 8,8 > O assigned. Under the hy- pothesis of the induction we may insert all the missing chains of dimension S p and have their diameters < 8. We thus obtain a new partial realization Q’’ of R,+41 in which only the (p + 1)-chains may be missing. But if Coss is any expected chain, the sum of the sets of its boundary chains is of diameter < a + 28 < 38. Therefore, if we choose 8B < 4 &(e, p), we may insert the miss- ing chains and choose them of diameter < «. Hence the condition of the theorem is fulfilled whenever the mesh a < n (3 é(e, p), p — 1) = ne, p).

14. THEOREM V. Given a compact metric HLC? space ® there exists for every e > 0 a@ quasi-complex V, into a subchain of which every c., gq S p, of N, is e-deformable over R.

Let {U*} be a f.c.0.s. of the HLC? space % of Theorem IV, and let a be the mesh and # the skeleton of the covering. We shall assume that {U‘} has the same skeleton ®, a choice always possible whatever a (see L1, p. 474). This means that any group of U’s intersect when and only when their closures do. As to the latter, we know that there exists a constant y < a such that whenever any set of diameter < y meets a group of U’s, these U’s intersect. From the preceding we conclude then that y has also the same property as regards the U’s themselves.

Let us mark on U‘ a point A‘ which we consider as an oriented zero-cell. If ®, is the sub-complex obtained after removing all simplexes of dimension > p from &, we may consider the set {A‘} as a partial realization ©) of a quasi- complex V, with the same incidence pattern as ®,. It may likewise be con- sidered as a partial realization ’ of a quasi-complex Y with the same pattern as®. From the construction of we conclude that the meshes of the two partial realizations do not exceed the maximum diameter 2a of the sum of any group of intersecting U’s. It follows in particular that if a < $n(8, p) we may complete VW; toa, of mesh < 8.

It is now a simple matter to show that if 6 = mesh R, < y then R, may be e-chain deformed into a quasi-complex whose elementary chains are subchains of V,. In fact the chain-deformation # may be so chosen that: (a) every ele- mentary ci of &, goes into (cj) times a vertex of &,; (b) if c} is an elementary chain of & with zero-chains in its boundary then # c} is the image in the trans- formation #, — VW, of a chain of ©, which is on the F(c) of the simplex o whose vertices have for images the points A‘ corresponding to the boundary zero- chains of c? ; (c) if c) is not of type (b) then 3 c} = 0.

CHAIN-DEFORMATIONS IN TOPOLOGY 13

We satisfy (a) by choosing for 8c} a chain (c})A* where A* is such that U* meets cj. By the same token (b) holds for g = 0, and (c) does not occur for that value. From this moment on the proof proceeds essentially like that of the deformation theorem of Topology, p. 92. The only point that may raise a question is the following. Granting that we have case (b) and that # F(c?), q < p, has already been described we must describe 8 c}. Owing to 6 < y the vertices A‘ to which # F(c}) belongs are on a set of intersecting U’s, and hence they are the images of those of aa of ®. It follows that 3 F(c}) is the image of a cycle T,-1 of F(c). Now o + F(c) has a subchain C, — I,_1, and the image of C, (under the chain-transformation @, — W,) is a chain c,‘ — F(c‘). We set vc = c, * and extend # to all g-chains by the linearity condition. The rest of the proof is as loc. cit.

15. Paraphrasing the treatment of L2, No. 5, we now define a set as strong HLC or merely HLC when it is HLC? for every p, and when in addition the function n(e, p) of Theorem IV has a lower bound n(e) > 0, independent of p.

The results of our paper regarding the comparison with retracts may also be extended here, provided that absolute retraction is defined relatively to imbed- ding not in an arbitrary set but in an arbitrary HLC set. We shall not stop to develop this point further. From Theorem V follows readily:

TueoreM VI. Every chain of an HLC is deformable into a subchain of a definite chain-complez.

For in the present instance we may complete VY, up to the dimension n of %, and have a complete chain-image VW of 6. We then apply the proof of Theorem V with the supplementary condition that dc} = 0 for q > n which is consistent since T,_1 = I’,, being now a chain of a c,, will be = 0. It follows that every R is deformable into a chain-complex whose elements of dimension p S n are subchains of ®, and the others are zero.

TuHeoreM VII. JfRis HLC* and n = dim §& is finite the space is HLC.

For § possesses no chains of dimension > n, hence in the construction con- nected with Theorem IV, one need never go beyond chains of dimension n and we may take for n(e) the least of the n + 1 positive numbers

n(€, p), Pp = O,1,--+,m.

16. THeoreM VIII. The homology groups of an HLC?, absolute or mod a closed HLC? subset, for dimensions S p, or of an HLC, absolute or mod a closed HLC subset, all have the same structure as for a finite complez.

By this we mean that they have finite bases with a finite number of relations between the elements of the bases, and furthermore for an HLC set that they are zero for dimensions above a certain integer n. In particular when the groups are free groups the Betti-numbers are all finite.

The proof is very simple. Take first the absolute groups and regular chains. The chains of 3 are the images in one of the correspondences $, — V,, ® — ¥

14 S. LEFSCHETZ

of certain chains of &, or @ and the correspondence preserves boundary rela- tions. Hence the groups {y,} of R (q S p for an HLC?) are isomorphic with certain subgroups of the same for ®, or ® respectively. They are therefore additive groups generated by a finite number of linear forms with coefficients in the abelian group M, corresponding say to chains y},i = 1, 2,---,7r. The homologies between the y’s are those on §, all of the form

(16.1) rny~0, rom. Since M is a field this system may be reduced by a change of base to the form (16 .2) yt'~0, i=1,2,---,r—8

and the first s elements of the new base form a minimum base for the g™ ho- mology group.

If we deal with singular chains, we may reason substantially as in the proof of the invariance of the combinatorial characters of a complex (Topology, p. 88) and show that (16.1) implies that there exists a chain of V,, or ¥ as the case may be,

(16.3) Cott > vi .

Since the y’s may after all include all the q-subcycles of ¥, or V, this shows that the qt homology group is the same as for Y, or for ¥ and hence the same as for , or for &, as the case may be.

17. Consider now a closed HLC? subset G of an HLC? space ® and let {U*} be the f.c.o.s. of No. 14. It may happen that certain sets U which meet G have an intersection which does not meet G. In that case their closure H also fails to meet G. Since the number of sets H/ is finite and they are closed, the distance 6 of their sum from Gis > 0. Let now, for every 7, V?‘ be the set of the points of U‘ nearer than 2/3 6 from G, and V?**' the set of those farther than 1/3 6from G. One of the two sets may well be vacuous. In any case however the non-vacuous sets V make up a new f.c.o.s. {V*} of 8, which like the initial covering has the same incidence pattern as the covering V‘ of the closures. Moreover, now if any aggregate of V’s intersect G, they intersect on G@ itself. Let us assume then that the initial f.c.o.s. {U‘} already possesses this property.

We now modify the construction of ¥, in No. 14, by choosing the point A‘ on G whenever U‘ meets G. Then, when the meshes are suitably small, we utilize the HLC? property of G to place on that set every chain of Y, whose vertices A‘ are on G. This is always possible since, after all, the f.c.o.s. {U‘-G} behaves relatively to G like {U‘} relatively to ® itself. As a consequence, 8 will transform a chain c; of & on G into chains of ¥, on G. Then, for suit- ably small meshes throughout, we shall be able to assign deformation-chains De; likewise on G. Under the circumstances the chains on G will be merely deformed over G and hence the cycles mod G will be deformed into cycles of Y, mod G. From this point on the rest of the proof is as for absolute cycles.

CHAIN-DEFORMATIONS IN TOPOLOGY 15

If # and G are both HLC the treatment is the same except that WV is to take the place of Y, throughout and the conclusion is again the same.

Corotuarigs. I. If Ris HLC’ and Gis HLC, q < p, the theorem regarding the homology groups of R mod G holds for dimensions S q.

Il. If n = dim & is finite and both R and G are HLC", the Betti-numbers abso- lute, or of R mod G, if any occur, are all finite.

AppuicaTION. If A isan HLC?-set, p S n, in an n-sphere H,, the Betti-numbers of H, — A are all finite; in particular Ro is finite, and hence H,, — A consists of a finite number of regions.

For dim A S n and the rest follows from the preceding results together with the extension of Alexander’s duality relation (Topology, p. 339).

18. HNR-sets. The closed set A shall be called an HNR of % whenever for every positive ¢ there is a positive 7 such that every chain c, within a spherical neighborhood S (A, n) of A is e-deformable onto A. These are obvious, but not complete, analogies of the NR property. The most important difference is that we have to use two neighborhoods of A, whereas in the NR there is only one.

THEOREM IX. Let R be compact, metric HLC. Then any closed subset A of R which is also HLC is an HNR of the space.

Let me), n2(€) be the constants of the HLC definition relative to ¢ and re- spectively to R and A. Let c, C S(A, 7») and let us impose upon it an ele- mentary decomposition of mesh making up a quasi-complex R. For each co of the subdivision we take a point B on A among those nearest to ¢. These points constitute a partial realization of mesh < 2 + £ of aR’ having the same structure as . By the HLC property of A, if we have a given a > 0 and if 2+ & < m(a) we may complete the partial realization to form R’ of mesh < a. Then &’ + & form a partial realization of a certain DR of mesh < yn + E + a, and if this quantity < £(e€) we may complete to form DR of mesh «. As a consequence §, and hence c,, will then have been e-deformed onto A. The two required conditions may be fulfilled by choosing

a<$m(e); &2< 432 (3 me).

Therefore the theorem holds, with the n(¢) of the HNR definition here chosen as § 2 (3 m(e)).

Remark. Here also as in Theorem II, the construction may be so modified that the elements of c,, or rather of its elementary decomposition, already C A are unmodified. For example, if F(c,) C A then it remains unmodified and

, Cy ™ Cp.

19. Relations between the classes LC”, LC and HLC”, HLC.

TueoreM X. In the system of singular chains an LC? is an HLC? and an LC is an HLC.

The proof is by means of Theorem I of L2 (p. 120) which is the analogue for LC-sets of our present Theorem IV. Assuming then ® to be LC? let &(e, p)

16 S. LEFSCHETZ

be the constant called n(e, p) in L2, p. 121 (proof of Theorem I), the change of notation being in order to avoid confusing that constant with the n(e, p) of Theorem IV. Let RK; be a partial realization of mesh < &(} ¢, p) of a certain R,. We may consider &, as the potential singular image of a certain simplicial complex K, and the chains of 8 correspond to certain chains of K whose sim- plexes make up a subeomplex K’ of K,. We may also consider R é as a singular image of certain chains and cycles of K’. By subdividing the chains of SR, and inserting suitable vertices on the chains already present we turn it into a singular image of the oriented simplexes which make up the chains in question in K. Now let cj be any expected chain of &, and let C} be the chain of K’ whose image, already present, is F(c}). We build up a new K, out of K’ by taking the join of some point P‘ with C} and adding this new simplicial chain to K’. We then proceed similarly by adding new two-chains for any c; whose boundary is now already represented in the complex already obtained, and so on until we have a new K, which may manifestly replace the former.

Now let c} be an expected chain of &, whose boundary is represented by certain chains in 2’ and which has necessitated the addition of a certain number of vertices P’ in building up the new K,. We represent these vertices by cer- tain points on F(c}), and if there are chains wholly unrepresented in the process we represent them by zero. We have now in the modified ’ a semi-singular image of K, in the sense of L2, and hence we may complete it to form a singular image on 8 whose mesh < ¢/3, and which contains a collection of chains making up a chain-image of &,. The chains are now singular with cells of diameters < §(€/3, p) < ¢/3, or else directly of diameters < ¢/3, and if c} is new, its singular cells meet the boundary chains of c}, already present which are on a set of diam- eter < &(€/3, p). Therefore mesh R < ¢€ and by Theorem IV, ® is HLC?, with n(e, p) = &(€/3, p).

If R is LC the constant £ has a positive lower bound &(¢) independent of p, hence n(e, p) 2 &(e) and Ris HLC. Our theorem is therefore proved.

20. Locally contractible spaces. According to Borsuk to whom this notion is due,'* a topological space ® is called locally contractible whenever for every

13 See K. Borsuk, Uber eine Klasse von lokal zusammenhangenden Réiumen, Fundamenta Mathematicae, vol. 19 (1932), pp. 220-242, notably p. 236.

Borsuk has shown (loc. cit., p. 240) that if dim R = vn is finite, we have in an obvious sense ANR~LC®. He then states as open (footnote 39) the question of the equivalence of the two properties when dim 9 is infinite. Now we have shown (L2, p. 121) that ANR ~ LC and, as we shall show by an example, LC ~ LC®, hence Borsuk’s question must be an- swered negatively. Thus our criterion for ANR is definitely the only one which holds independently of the dimension. This emphasizes once iore the importance of the semi- singular complex introduced in L2, and of its analogue, t::e partial realization of chain- complexes of the present paper.

The example alluded to above was communicated to us by Borsuk and is as follows. Consider in the Hilbert parallelotope a sequence of points {z,} on a segment Oz, from the origin O, — O monotonely, and let H, be a euclidean p-sphere of center z, and radius

CHAIN-DEFORMATIONS IN TOPOLOGY 17

open set U containing any point z there is another, V > U, which is homotopic to a point on U. Let ® be compact, metric, locally contractible and consider the class of all spheroids {S(z, e)} of R with fixed radius «. For every point z there is a spheroid S(z, £) which is e-homotopic to a point. As the space ® is compact it has a f.c.o.s. {S(a;, £)} and there is a constant » such that every subset of #% whose diameter < 7 is on one of the spheroids S(z;, £) and hence e-homotopic to a point on §.

It follows immediately from what precedes that given ¢ every singular p- sphere of St whose diameter < » bounds a singular (p + 1)-cell of diameter < «, where n(e) is independent of p. We might call this property weak LC.“ Ow- ing to Theorem X this implies that R is a weak HLC in the system of singular chains. That it isa weak HLC in any system is readily shown as follows. Any cycle y, such that diam | y, | < 7 is e-chain deformable, by virtue of Theorem I, into a point, which is a y, ~ 0, when p > 0, and which is taken (yo) times when p = 0. Therefore (No. 11) ® is p-HLC for every p, and since we have been able to choose n(¢) independently of p, we have

TuHeoreM XI. A locally contractible compact metric space is both a weak LC and a weak HLC.

While local contractibility is not sufficient to insure the LC or HLC proper- ties, this will be the case if we strengthen it in the following manner. Let us call the open set U self-contractible whenever it is homotopic to a point on itself, i.e., whenever it is its own set V in the definition of local contractibility. A class of open sets {U} will be called self-contractible whenever it consists of self- contractible sets and is closed with respect to intersection (if U’, U” are in the class so is U’-U”). A compact metric space R which possesses for every ¢ a self- contractible class whose sets are all of diameters < «, is both strong LC and strong HLC. The proof is essentially parailel to the analogue in L2, pp. 126, 127 (No. 17), and need not be repeated here.

21. Application to locally polyhedral spaces. We shall say that ® 1s locally polyhedral whenever every point x has a neighborhood whose closure is a finite complex K*. We may replace K* by a subdivision with z as a vertex and then the neighborhood by the star of x in K*. Therefore ® may be defined as a space of which every point has for neighborhood the star of a vertex in a finite complex. Since every star is homotopic on itself to its own vertex, the space §t is locally contractible and it is also clearly locally compact.

Suppose now §t to be also compact and metric. Since it may then be covered

< 3 d(xp, Zp-1 + Zpti). MR is the sum of the spheres + O + the segment with the open diameters of the spheres on the segment removed. It is readily shown that the p-LC condition holds for every « with the corresponding n(e, p) < diam Hy. Hence n(e, p) ~ 0 with increasing p and ® is LC® but not LC.

14 Ts the converse true, i.e., is every weak LC locally contractible? Is weak LC ~ LC? For the present these questions must remain unanswered.

18 S. LEFSCHETZ

with a finite number of the stars, its dimension is finite. Combining this with No. 17 Corollary II and Theorem XI, we have then

THeoreEM XII. A locally polyhedral compact metric space is both LC and HLC.

Corotuary. The homology groups of the space of Theorem XI, absolute or mod a closed subset of same type, have the structure of those of a finite complex. In particular its Betti-numbers, if any occur, are also finite.

IMPORTANT SPECIAL CASE: The space and its subset are absolute topological manifolds.

Of course in both theorem and corollary the space is assumed compact.

22. Extension to locally compact separable spaces. Substantially all the results of the present section may also be extended to these spaces. We recall that any locally compact separable space is also metric, although the reverse need not hold.” As far as our definitions are concerned, the HLC? or HLC conditions formulated in No. 11 in terms of ¢, 7 must now refer to the cycles meeting a specific self-compact subset A of ®, and must hold for every such A. However n(A, ¢, p) or n(A, €) may well vary with A. The U, V con- ditions applied to every point of A may be replaced in fact by e¢, » conditions on the chains meeting A, but not on all chains. Keeping this in mind we verify at once that Theorem IV holds when the elementary chains of &’ all meet A. In Theorem V the chains considered must now all be chains of A and the def- ormation is on any preassigned open set U > A. The subsequent results on the homology groups hold regarding the homologies between chains of A on U, that is to say, when bounding is allowed not merely on A butalsoonU. Theorem X may be derived, provided that we extend, as we may readily do, Theorem I of L2 to locally compact separable spaces. It is always understood that the semi- singular complex loc. cit. must have all its (realized) elements meeting A. Simi- larly Theorem XI holds also, it being always understood that the constants e, are related to A as described above. Since Theorem XII is a mere corollary of Theorem XI, it is likewise applicable to locally compact separable spaces.

23. Concluding remarks. The theory developed in the preceding pages was not intended to be exhaustive, but only to cover enough ground for some fairly immediate applications. Thus by means of quasi-complexes mod A, we could introduce chain-deformations mod A and extend in an obvious manner the results of the present paper. As already observed, another extension would be to non-compact spaces, say to the very general spaces whose homology theory has been developed in recent years by Cech. These extensions however would be strictly mechanical and may safely be left to any experienced reader.

Princeton, N. J.

15 See Alexandroff-Urysohn, Mémoire sur les espaces topologiques compacts, Verhandelin- gen der Koninklijke Akademie van Wetenschappen te Amsterdam, afdeeling natuurkunde, deel XIV, No. 1 (1929), p. 83.

ON CONVERGENCE IN LENGTH By C. R. Apams anp Hans Lewy

1. Introduction. Adams and Clarkson! have recently considered a sequence of functions f,(x) (n = 1, 2, 3, ---) defined on an interval* (a, 6) and subject to the following conditions: (i) f,(2) tends to a limit function fo(x) of bounded variation; (ii) the total variation T°(f,) of f,(x) on (a, b) tends to the total variation T°(fo) of fo(z) on (a, 6). The notation f,(z) — v — fo(x) was em- ployed to describe this situation, the symbol — v — being read ‘“‘converges in variation.’”’ Under these conditions each curve y = f,(x) for n sufficiently large has a finite length in the sense of Peano; and the pair of conditions, (i) and (ii’) L2(f,) — L®(fo), define what may be called “convergence in length”, the notation f,(x) — 1 — fo(x) being used for brevity.‘

The reader of AC may very naturally raise a question as to the relation between convergence in variation and convergence in length; to compare these notions is one of the objects of the present note.

Definition of notation. It will be convenient usually to designate by S, with or without a subscript, a set of points a = xo, 11, --- , 2» = 6, with

Ze < my < ++- < dp

In general B will stand for a broken line inscribed in a curve y = f(x) and con- sisting of p segments, the 7“ segment (¢ = 1, 2, --- , p) having end-points at (xia, f(zi-1)), (xi, f(z,)) and length 6; The function whose graph is B will frequently be referred to as B(x). Broken lines inscribed in two distinct curves and determined by the same set S will be called “‘corresponding”’.*

Received February 8, 1935; presented to the American Mathematical Society, February 23, 1935.

1C. R. Adams and J. A. Clarkson, On convergence in variation, Bulletin of the American Mathematical Society, vol. 40 (1934), pp. 413-417; this paper will be referred to as AC.

2 The closed interval is always to be understood.

3 According to Peano, a curve (continuous or not) is said to have finite length if the lengths of all inscribed broken lines B are bounded; by definition their least upper bound is the length of the curve.

4S. Bochner has imposed the condition of convergence in variation (‘‘abgeschlossene Konvergenz’’) to permit passage to the limit in a Stieltjes integral: Monotone Funktionen, Stieltjessche Integrale und harmonische Analyse, Mathematische Annalen, vol. 108 (1933), pp. 378-410. This condition and that of convergence in length are sometimes employed in the calculus of variations; see Tonelli, Fondamenti di Calcolo delle Variazioni, vol. 1, Bologna, 1922.

5 Incidentally we shall nowhere employ the fact that an absolutely continuous function is the integral of its own derivative. On the other hand, come of the results herein con- tained can be used as a basis for the deduction of theorems concerning summable functions; for instance, the possibility of approximating in the mean a summable function by step- functions may be concluded from Theorem 4, and the fact that an absolutely continuous function is the integral of a summable function follows easily.

19

20 Cc. R. ADAMS AND HANS LEWY

2. Preliminary remarks. Since L and T for a function f are defined re-

spectively as the least upper bound of LZ and of T for broken lines inscribed in f, we have at once the fundamental semi-continuity relations: that f,(2) — fo(x) on (a, b) implies (1) lim Lo(f.) = Li(fo) lim T3(fn) = T2(fo). Hence the relation f,(z) — 1 — fo(x) [fn(xz) — v — fo(x)] on (a, b) implies that relation for every subinterval. Moreover it may be observed that the relations f.(x) — fo(x) on a set of points everywhere dense in (a, 6) and L°(f,) > L°(fo) [T° (fn) — T2(fo)], when fo(x) is continuous on (a, b), imply f,(x) — 1 > fo(x) [fn(x) — v — fo(x)] on (a, b).

If f(x) is of bounded variation on (a, b), we have (2) b—a+ Tif) = Lilf) = {6 — a)? + (TAP),

both of the inequalities flowing at once from the definitions of Z and T. From (2) it follows (i) that each of the relations f,(x) — l— c, f»(x) — vc (e = con- stant) on (a, b) implies the other; and (ii) that L7(f) is continuous on the left [right] at each point x where f has this property. If f(x) is absolutely contin- uous on (a, b), both TZ(f) and Lz(f) are likewise.

Whether f (of bounded variation on (a, b)) is continuous or not, there always exists a sequence of inscribed broken lines B, (n = 1, 2, 3, ---) such that we have both B, —l1—fand B, —v—f. For the set S, determining B, can be

so chosen that S,4; includes S, for each n, and S = & S, is everywhere n=1

dense in (a, b) and contains all the points of discontinuity of f. Then B,(zr)

converges to f(x) on (a, b): at each point ~’ in S the convergence is evident,

while at a point ~ not in S we have

lim | f(§) — Ba() | S TQ) + lim | f(’) — Ba(@’) | + lim T}(B,)

< 2TH ()

for t’ < § < £” and &’, &” in S;since f is continuous at £, the right-hand member tends to zero with ¢”” — &’. From (1) and the definitions of L and T follow the relations L°(B,) — L°>(f) and T°(B,) — T°(f).

3. We first prove

THEOREM 1. The relation f,(x) — l1— fo(x) on (a, b) implies f,(x) — v — fo(x); the converse is not true even when f,(x) (n = 0, 1, 2, ---) ts assumed absolutely continuous.

Any e(> 0) being given, let B be a broken line inscribed in fo and satisfying the inequality L°(B) > L°(fo) — «. For convenience we set

ae (fo) — b = &, ie— %u = d;;

ON CONVERGENCE IN LENGTH 21

then = ¢; is clearly < ¢ and there exists an m such that we have Ls‘_, U) & Li'_, Wo +0 (i =1,2,---,p;n>m). From (2) we thus obtain for each 7 T:'_,U) ${[Ls‘_, mi - a" S [(bi + 2e:)* — di}? (bt — d?)\® 4 2(be)!” + 2a,

and hence, by aid of Schwarz’s inequality,

IIA

p Pp

Tf.) $ T°(f) +23 be «) 42 Ss

1 i=l {=}

Ta(fo) + 2 [eLa(fo)]'!? + 2c.

In view of (1) the direct statement in our theorem is now proved. An example illustrating the failure of the converse, even when all the functions involved are absolutely continuous, is provided by the sequence f,(x) defined in §5 of AC. Corotitary.S L® [f(x)] = L? [T?(f)]. From (2) it follows that neither or both of these quantities are finite. In the second case the relation

| f(z) — f(@ia) | S TA) — THY)

shows that, of two corresponding broken lines inscribed in f(x) and Ti(f), the former has length no greater than the latter. Hence L°[f(x)]is < L°[T2(f)]. On the other hand, for any broken line B we have L°(B) = L®[T2(B)]; hence, choosing a sequence of broken lines B, (n = 1, 2, 3, ---) with B, —l—f we obtain

:

IIA

Li(f) = lim L2(B,) = lima Le (T<(B.)] 2 La(Tahl,

since by Theorem 1 77(B,) tends to T7(f).

By virtue of Theorem 1 and the results obtained in §3 of AC, and by reason- ing analogous to that employed in §3 of AC, we may readily establish the follow- ing theorems.

TuHeoreM 2. The relations f,(x) — 1— fo(x) on (a, b) and f,(z’) = f(x’ — 0) [fo(x’) = fo(x’ + 0)] imply that x’ is a point of uniform convergence on the left [right] for f.(x), To(fn), and Le (fn).

Corotuary. The relation f,(x) — 1 — fo(x) on (a, b), with fo(x) continuous, implies that the convergence of f,(x) to fo(x), of T2(f,) to T2(fo), and of L2(f,) to Li (fo) is uniform over (a, b).

THEOREM 3. The relations f,(x) — l—fo(x) on (a, b) and fo(x’) ¥ fo(x’ — 0)

6 Dr. J. A. Clarkson suggested this relation and gave a proof.

22 Cc. R. ADAMS AND HANS LEWY

[fo(a’) # fo(x’ + 0)] imply that x’ is a point of uniform convergence on the left [right] for f.(x), Tz (fn), and L2(f,) or for none of the three, according as f,(x' — 0) [fn(x’ + 0)] tends to fo(x’ — 0) [fo(x’ + 0)] or not.

Corotuary. If we have f,(x) — | — fo(x) on (a, b), a necessary and sufficient

condition that T(f,) — Ti(fo) uniformly on (a, b), or that L2(f,) — L2(fo) unt- formly on (a, b), is that f,(x) — fo(x) uniformly on (a, b).

4. Further inequalities. If f(x) and g(x) are of bounded variation on (a, b), we have

(3) To(f) + Tag) = Ta(f + 9);

(4) Li(f) + Teg) = Lilf + 9);

(5) M:T3(f) + MiT2(g) = T2(f-9);

where M, = L.u.b. | f | and Mz = Lub. | g| ;

(6) [Li(nP — (6 — a)? (1 + m’) = [TE(f — max)P/(1 + m’),

where m = [f(b) — f(a)]/(b — a).

Relation (3) is well known and obvious; (5) is an immediate consequence of the definition of T. Relation (4) follows at once from the fact that the 7 seg- ment of a broken line inscribed in f + g has length no greater than the cor- responding segment inscribed in f plus the quantity | g(a:) — g(ai-1) |.

Proof of (6). We first establish the inequality for the case of a broken line B, denoting the derivative B’(x) by ¢(x) so that B(x) = [ ¢g dx. Then (6)

a reduces to’

@ | fa +enirdr] "2 (b — a)*(1 + m?) + | [ie —m| ax] + m?),

b where m = i ¢gdx/(b — a). Let us designate by / | / | an integral over

+

the set of points x for which g — mis 2 0[< 0]. By Minkowski’s inequality

we have + a +

together with the same relation for [ Setting

a= | ar, p= | ede, y= | a, 6= [ eae,

7 It is of interest to note that, as soon as (6) is proved, (7) holds for any summable func- tion ¢.

ON CONVERGENCE IN LENGTH 23 we obtain m = (8 + 6)/(@ + y) and

[ie-miac= [max — [ (e — mas

2(By — ad)/(a + ),

and (7) will be true if

4(By = ad)? | 1/2

(a + y)? + (8 + 4)?

holds. But under a suitable orthogonal transformation of a £, n-plane, the two

points £ = a,» = Band — = y, » = 6 respectively go into (a’, 8’) and (7’, 6’)

with 6’ + 6’ = 0, while (8) is invariant and reduces to the triangular inequality | a’ + ip’| + | 7’ — i8’| = | a’ + 7’ + 2%8’ |.

Relation (7) having now been proved, (6) follows at once by approximating f

by a sequence of inscribed broken lines B, (n = 1, 2, 3, ---) with B, —l—f and taking account of (1).

(8) |a+iB|+| 7+ i) z [+t erat

5. Addition and multiplication of sequences. In this section we examine the question of invariance under addition and multiplication of the prop- erty of convergence in length. That convergence in variation is not invariant under these operations, even when all the functions involved are assumed absolutely continuous, has been shown in AC. That convergence in length is not invariant under addition, even when the limit functions are assumed continuous and all the approximating functions absolutely continuous, may be seen from the following example.

Let fo(x) be defined on (0, 1) as the Cantor function, continuous but not abso- lutely continuous,’ and let fo(z) = 1 for 1 S x S 2. We first observe that, if B is any broken line inscribed in fy on (0, 1), we have

Ti (fo — B) = [ | B’ | dx + Th(fo) = 2;

this shows that f, — 1— fo on (0, 1), with fo continuous and f, (n = 1, 2, 3, --+) absolutely continuous, does not imply T}(f, — fo) 0. In view of a theorem of Plessner® there exists a 8 > 0 and a sequence of positive numbers a, (n = 1, 2, 3, ---) such that a, > 0 as n— © and T}[fo(x) — fo(z + a,)] is > B for each n. Let a sequence of sets S, (n = 1, 2,3, ---) on (0, 1) be chosen as

§ Any other function of this character will do equally well.

® Plessner, Eine Kennzeichnung der totalstetigen Funktionen, Crelle’s Journal, vol. 160 (1929), pp. 26-32; see also Dunford, On a theorem of Plessner, to appear in the Bulletin of the American Mathematical Society, vol. 41 (1935).

24 , C. R. ADAMS AND HANS LEWY

follows: for each n let the set S, determine a sum approximating T} [fo(x) — fo(x + a@,)] within 8/2; let sequence S, be selected so that S,

includes S,_, + S;, for each n and S = = S, is everywhere dense in (0, 1). n=1

Designate by B,, B*, and C, respectively the broken lines inscribed in fo(z),

fo(z + en), and fi(x) — fo(a + a@,) and determined by S,. We then have

B,(x) — B(x) = C,(2), BY(x) — fo(z) = B(x) — Cr(x) — fo(x), Ba(x) — fol),

and | C,(x) | S max | fo(x) — fo(x + an) | — 0, whence B*(xr) — fo(x). More-

over we have by (1)

Lj [fo(x)] S$ lim Lj (BX) S lim L5(B%) S lim Lj [fo(z + an)] = Lo [folz)].

no no

Hence we have B® —1-— fy. We also have B, —l1—>fo. On the other hand T)}(B, — B*t) = T}(C,) is > 8/2 for each n, which implies (see Theorem 1) that B,(x) — B*(x) does not converge in length to the function zero.

If, however, the limit functions are absolutely continuous, the property of convergence in length is invariant under addition and multiplication inde- pendently of the nature of the approximating functions. To establish this fact we first prove”

TueoreM 4. The relation f,(x) — 1 — fo(x) on (a, b), when fo(x) is absolutely continuous, implies T°(f, — fo) > 0.

That the hypothesis of absolute continuity for fo cannot be dispensed with is shown by the example just described above. For the proof of the theorem we employ two lemmas concerning an absolutely continuous function fo, as follows.

a) If fo is absolutely continuous, S an arbitrary set subdividing (a, b), Bo the broken line inscribed in fy and determined by S, and 3%), (h = 1, 2, --- , q) those subintervals in which the slope m, of Bo is in absolute value = M, there exists a function a(M) independent of S such that

> T3,(fo) S a(M), and a(M) —0 as M-—~o. h

To prove this assertion, we observe first [3,M S 23; | m,| S T?(fo), whence >9, S T?(fo)/M. Then, on account of the absolute continuity of fo, TZ (fo) is absolutely continuous and for any set of non-overlapping intervals the sum of whose lengths is < T°(fo)/M, and in particular for our set of intervals &;,

we have =T3,(fo) S a(M), with a(M) — 0as M— ~. b) The hypotheses of a), together with f,(2) — v — fo(x) on (a, 6), imply lim = T3,(fn — fo) S lim = Ty(fn) + 2 Tax(So) S 2a(M).

no h

1” A proof has been given by Tonelli under the additional assumption that all the func- tions f,(x) (n = 1, 2, 3, ---) are absolutely continuous; loc. cit., pp. 186-187.

ON CONVERGENCE IN LENGTH 25

The conclusion is immediate, since only a finite number of intervals %, is involved and in each we have T(f,) — T(f).

Turning now to the proof of Theorem 4, let S be any set on (a, b) determining a broken line B,, inscribed in f, (n = 0, 1, 2, ---); for each subinterval defined by S we clearly have

lim T7!_ (fn — fo) S lim T7‘_, (fn — Bn) + lim T;'_, (Bn — Bo)

ne no

+ T.\_(fo — Bo) < lim Ty‘, (fn — Bn) + Tz'_, (fo — Bo).

no

For those intervals 3, for which the slope m, of By is numerically 2 M, we have by Theorem 1 and lemma b)

(9) lim = T3,(fa — fo) S$ 2a(M), a(M) > 0as M— o,

no

Denoting by 2’ the summation over only those intervals of S for which the slope m, of By is numerically < M, we obtain by (6) and Schwarz’s inequality

ee (fo — Bo) S$ (1 + M?)!/2 37 | [eet (fo) |? - tH ea"

(10) S14 MY"PE[ LEi_ (fo) + LEt_, (Bo) || L2t_o) — L2{_,(Bo) |" < (1 + M*)'*K(Bo), where K(Br) = (L2(fo) + L2(Bo)}'? (L2(fo) — Li (Bo).

Similarly, on summing over these same intervals, we find in view of the con- vergence of B, to Bo

lim 2’7, ' i Gn — Bn) S (1 + M?)"? lim [L2 (fn) + L2(B,))?(L2(f.) — L2(B,))"2.

Hence from L°(f,) — L?(fo) and L°(B,) — L°(Bo) we infer (11) lim 2T,' : -1 fn — B,) S (1 + M*)'?K(B,).

From (9), (10), and (11) we obtain lim T°(f, — fo) S 2(1 + M?)'/2K(Bo) + 2 a(M).

This relation is independent of S; a(M) can be made arbitrarily small by taking M sufficiently large and, M having been fixed, the first term on the right can be made arbitrarily small by a suitable choice of S. Consequently we have lim T°(f, — fo) = 0, and the proof of Theorem 4 is complete.

no

26 C. R. ADAMS AND HANS LEWY

TueoreM 5. The relations f,(x) — fo(x) on (a, b) and T2(f, — fo) — 0, with fo(x) of bounded variation, imply f,(x) — 1 — fo(x). For by (1) and (4) we have

Lo(fo) = lim Lo(f.) S lim Le(fn) S Le(fo) + lim Tafa — foc) = La(fo).

no noe n—72

THEOREM 6. The relations f,(x) — 1 — fo(x) and g,(x) — l — go(x) on (a, b), with fo(x) and go(x) absolutely continuous, imply both

[fn(z) + gn(x)] — L—> [folx) + go(x)] and fr(x)gn(x) — 1 — fo(x)go(z). From (3), (5), and Theorem 4 we infer T2(fn + gn — fo — go) S Ta(fn — fo) + T2(Gn — go) — 0, T3(Sngn — Jogo) S M-Ti(fn — fo) + N-T2(gn — go) — 9,

M being a uniform upper bound for | g, | and N an upper bound for | fy |. The conclusions then follow from Theorem 5.

Brown UNIVERSITY.

ABELIAN SUBGROUPS OF ORDER p”" OF THE I-GROUPS OF THE ABELIAN GROUPS OF ORDER p" AND TYPE 1, 1, 1, ---

By HENRIETTA TERRY

The group of isomorphisms J of a given group is important in the construction of new groups which contain the given group as an invariant subgroup. How- ever, little is known about groups of isomorphisms in general, and even when the given group is of the simplest type, i.e., abelian, the group of isomorphisms has not been thoroughly studied except in a few extremely special cases.

For instance, the J-group of the cyclic group has been discussed by Burnside! and Miller.2. The J-group of a group of order p" and type n — 1, 1 was studied by Miller? and although the general type has been barely touched upon he has proved several general theorems. The J-group of the abelian group H of order p”" and type 1, 1, 1, --- was considered by Moore,’ and it is well known that the operators U of this group can be represented as non-singular linear trans- formations on the exponents of n independent generators of H. Thus a deter- mination of the subgroups of the group of these n-ary linear homogeneous trans- formations modulo p is equivalent to a determination of the subgroups of the I-group of H. Dickson determined all the subgroups in the case n = 3,° and the subgroups of order a multiple of p in the case n = 4,’ and all the subgroups of the three highest powers of p for all positive integral values of n.°

A necessary and sufficient condition that an operator U in J be of order a power of p is that the characteristic determinant of U be (— 1)"(A — 1)"° The invariant factors of such operators are powers of (1 — A) which determine the canonical form and conjugates of U. There is a one-to-one correspondence between these powers of (1 — \), the conjugate sets of operators in J whose orders are powers of p, and the partitions of n. Hence, we shall designate an operator U of I of order p™ by the degrees of its invariant factors or by the partition of n to which it corresponds, i.e.,

n= Mm + Ne + M3 + +++ $ Ny + My 41 $ ees + Ms,

where the terms are ordered so that n; = nis, ny > 1, ny4i = L,ify < 6.

Received by the Editors of the American Journal of Mathematics, September 15, 1934, accepted by them, and later transferred to this journal.

1 W. Burnside, Theory of Groups of a Finite Order, 1897, pp. 239-242.

2G. A. Miller, Transactions of the Amer. Math. Soc., vol. 4 (1903), pp. 152-160.

3G. A. Miller, Transactions of the Amer. Math. Soc., vol. 2 (1901), pp. 259-264.

*G. A. Miller, Annals of Mathematics, (2), vol. 3 (1902), pp. 183-184; Amer. Journ. of Math., vol. 36 (1914), pp. 47-52.

5 E. H. Moore, Bulletin of the Amer. Math. Soc., vol. 2 (1895), pp. 33-43.

6 L. E. Dickson, Amer. Journ. of Math., vol. 27 (1905), pp. 189-202.

7 L. E. Dickson, Amer. Journ. of Math., vol. 28 (1906), pp. 1-16.

8 L. E. Dickson, Quarterly Journ. of Math., vol. 36 (1904-05).

* H.R. Brahana, Proc. of the Nat. Acad. of Sci., vol. 18 (1932), p. 724.

27

28 HENRIETTA TERRY

Generators of H can be so chosen that U can be written as a set of partial transformations each on n;, i = 1, 2, 3, --- , 6, generators distinct from those transformed by the other 6 — 1 partial transformations.” Dickson designates the independent generators which are obtained from one of these partial transformations as a chain of length n;. In the following pages we shall refer to these partial transformations as chains of length n; in U.

The partitions of n are ordered so that the partition

mtmt+:---t+tn+14+14+14--- is greater than the partition mtr tere $n, +l+1l+ilt---, if (1) m+ m+ --- +n, Sn, +g +--+ Ni0 (2) n. = n; and (3) at least one nj; > n;. For example, of the two partitions a+b+4+14+14---

and

o4+6423434141+4...

the first is the greater. This ordering of the partitions of n makes it possible to characterize simply a set of abelian subgroups of order p™ of I whose conjugates contain every abelian subgroup of order p* of J.

The operators in J of the type

1 diz a3 ai4 s+ Gin 0 1 23 Ao Aan (1) 0 0 1 a34 A3n 0 0 0 1 Aan 0 0 0 0 see 1

form a group which is a Sylow subgroup of J. We shall designate (1) by J, in the following pages. The maximal abelian subgroups of J,, in which every operator corresponds to a partition of k or less chains of length 2, have been determined."

We proceed to obtain the form and the order of the maximal abelian subgroups of J, in which the greatest partition to which any operator in the group cor- responds is m + m2 + --- +n, +1+1-+41+4-.---. The groups are deter- mined throughout by obtaining all of the operators of J, which are commutative

10 L, E. Dickson, Modern Algebraic Theories, 1926, p. 90. 1H. R. Brahana, On metabelian groups, Amer. Journ. of Math., vol. 51 (1934), pp. 490-510.

ABELIAN SUBGROUPS OF ISOMORPHISMS 29

with an element U, of the type and which with U; do not generate an operator corresponding to a greater partition. The form chosen for U, corresponding to the partition 3 + 1+ 14 --- is

1 1 0 QO <«- @Q

0 1 1 . we” @

U,=1|0 0 1 0 0

0 0 0 S ssc. 9 The operators in 7, commutative with U; uniquely determine the form of the maximal abelian subgroup except for chains on S;, --- , S,. Obviously U, and any operator not commutative with S;, --- , S, will generate an operator

corresponding to a larger partition. Given any partition to which an operator is to correspond a simple form analogous to U; can be written.

Operators corresponding to partitions which have k equal terms different from 1 are considered first. The maximal abelian subgroup corresponding to the partition 4 + 4+4+1+1+414 --- which follows indicates the form for the general case of k chains of length n.

Type III, l a2 a4 OF O O O O DO Ayjro Gia Qija2 +++ Gin | 0 1 az. 0 O 0 0 0a O O --- O 0 0 1 0 0 0 0 0 0 ae 0 O --- O 0 0 O 1 ads ae O ODO O Agro Agar Ayre +++ Gin 0 0 0 0 1 a 0 0 0 O ag O --- O 7; 6 8 8 tt @ 8S @ Oa OC «ss G 0 0 0 0 O O 1 Azg Az9 7,10 A711 A732 +++ Az,n! 0 0 0 0 0 0 0 1 ag 0 O ayy 0 | 0 0 0 0 0 0 0 0 1 0 O ay --. O 0 0 0 0 0 0 0 0 0 1 90 0 0 | a 0 0 0 0 0 0 0 0 0 0 0 0 «+. 1)

Since there are n — 7 independent a’s to be determined in each of the 3 chains the order of the group is p*““~”. The results and generalizations for k chains of equal length are tabulated below.

30 HENRIETTA TERRY

Type Order of Maximal Generalizations Abelian Subgroup I, p" I; p.- I,, p I, gt II. prn-2) II; prn-9) II, prn-ny Il, pin» III, pin-s) III; prn—s) III, prin—2n, +1) III, pin IV; pin IV; pin? IV,, piin—3n, +2) IV, piin—10) Ke pra» prla— hn, +G—-2)] K; prin—2k+1) K,, pr ln—k(m 1) +(m,-2)]

K, gPa-Se pr-rtsy—-D

The operators corresponding to partitions whose terms different from 1 are unequal were studied next and the form of the maximal abelian subgroup of J, which corresponds to the general partition

e+ eet ++ +e 4+1341414->-

determined. This includes the groups corresponding to partitions of n having k equal terms, described in the preceding pages, and a check on the following results is obtained by substituting equal n; and comparing with the results given in the table.

In the first row of the chain of length n; there are n; — 1 zeros for each follow- ing chain while the second, third, --- , and (m; — 1)** rows have only de- pendent a’s. Likewise the first row of the chain of length nz has n; — 1 zeros for all following chains and the remaining rows have only dependent a’s. The first row of the chain of length n; has n. — 1 zeros due to the chain of length ne, but the first row of the chain of length nz has n,; — 1 zeros to the left of the main diagonal due to the chain of length n;. Hence the number of independent a’s in the first row of the chain of length mn is n; — nz less than the number of inde- pendent a’s in the first row of the chain of length m. This can be extended readily to n; for all ¢.

It follows that the order of the group is p with the exponent a, + --- + a, where aj = n — [1+ m—1+n;—1+--- +n, —1+ (m —1n,)]. Col- lecting exponents we have

prr-lr—-D(aytayt ‘$n, )—7(y—2)] -

or

pr-rtsr-D

ABELIAN SUBGROUPS OF ISOMORPHISMS 31

Every subgroup of order p", m <n, is contained in at least one Sylow sub- group of order p", hence the groups determined are maximal abelian subgroups of order p™ of I of H.

Furthermore, the operators in J of order a power of p corresponding to a given partition are conjugate, hence each is in the maximal abelian subgroup cor- responding to its partition determined above or one of its conjugates in J. Moreover, every abelian subgroup of order p” in J is conjugate to an abelian subgroup of J,. Consequently it is in a conjugate of the maximal abelian sub- group corresponding to the greatest partition to which any operator in the sub- group corresponds. Therefore the abelian subgroups of order p”™ in J are completely determined when the abelian subgroups of these maximal abelian subgroups are characterized.

In order to accomplish this task for subgroups of order p? it is expedient to consider the groups of order p"*? which contain H as a maximal abelian in- variant subgroup. Miller’? determined all the groups of order p"*'!, p being any prime, which contain the abelian group of order p" and of type 1, 1, 1, --- .

We proceed to determine the groups G, of order p"*? which contain the abelian group H of order p” and of type 1, 1, 1, --- invariantly, generated by H and two independent commutative operators of order p in J corresponding to small partitions of n. In the groups first considered the operator in {U,, U2} cor- responding to the greatest partition has only one chain and, if this operator has a chain of length greater than 2, there is at least one operator, hence a subgroup of order p, in {U,, U2} corresponding to a lesser partition.

Moreover, two commutative operators of J of given chain length can not generate an operator with a chain of greater length than either, although some operator in the group generated by them may contain more chains than either. From this and the above we have the following

THEOREM 1. If a group of order p? is generated by two operators of I with chains of length n, and ne, respectively, on n, independent generators of H, n, > ne, then (p — 1) of the operators have a chain of length nz and the remaining ones, except the identity, a chain of length n,.

Using this theorem we determine the number of abstractly distinet groups of order p"*?, of class 3, and with central" of order p"~* containing H as a maximal abelian invariant subgroup, extended by 2 commutative operators of order p in J.

In order to have at least one operator with a chain of length 3, we shall de- fine U, by

Uy SU, = S,So, U7'S.U, = S283, and let U2 satisfy the relations

U;z'S,U2 = S,Si, Uz'S.U2 = S283.

12 G. A. Miller, Bulletin of Amer. Math. Soc., vol. 8 (1901-02), pp. 391-396.

13 The class of G; is equal to n; of U:.

14 The order of the central of G, is equal to the number of terms in the partition to which U, corresponds.

32 HENRIETTA TERRY

The independent generators of H invariant under U; and U2 can be renamed so that S, may have the form S, = S3S3S}7, and by Theorem 1 @ can be made zero, hence the most complex form of S; which we need to consider is 8, = S3S{ as the exponent of S; will be 1 for some power of U2.

When a = 0, we have a group abstractly distinct from the case a ¥ 0, as in the first case the commutator subgroup K of G2 is of order p?, while in the second it is of order p*. The condition a = 0 uniquely determines U2 and the sub- stitution S; = S;S% changes an operator with the exponent of S; not equal to zero to one in which it is zero. This concludes the consideration of G,; = {H, U,, Us}, where U; and Uz are of type Is, or less, with one group for K of order p*, and one for K of order p?, each of which has one subgroup of type I».

Further application of Theorem 1 shows that for the groups of order p**? of class 4 and central of order p"~* obtained as before there are four distinct groups as follows:

Order of K No. of groups No. of chains of length 2 in U2 p* 2 1,2 p* 2 3 3.

And for a group of order p"*? and class n; with a central of order p*~":*! con- taining H as a maximal invariant abelian subgroup generated by H, Ui, U2, U, and U; being commutative operators of order p and U, having one or two chains of length 2, we have the following:

Order of K No. of groups No. of chains of length 2 in U2 p™ 2 1,2 pa! 2 1,2.

At this point we call attention to the kind of operators which appear in the maximal abelian subgroup of type I,,:

1 de as Ge ss Ain © Aint t+ Gin 0 1 G2 M3 ++ Gynt 0 «>» O 0 0 lL aye +++ Gane OO

0 0 0 1 +++ Gin 0 s+» 60

0 0 0 aye 0 0 0 0 1 0 0 0 0 0 0 a |

If a2 = G3 = --- = G,n,-1 = 0, the operator is of type kh.

ABELIAN SUBGROUPS OF ISOMORPHISMS 33

If ay. = G3 = +++ = 4),x,-2 = 0, the operator is of type IIs. If ay. = a3 = +--+ = j,n,-3 = 0, the operator is of type III».

If ay. = is = +--+ = Qi,n,-x = 0, the operator is of type Ky, where k S n,/2.

If aye = 0, a;3 + 0, there are two cases: (1) m = 2c —1 One chain of length c and one chain of length c — 1. (2) nm = 2c Two chains of length c.

If ay. = a3 = 0, ay * 0, there are 3 cases: (1) m = 3c — 2 One chain of length c and two chains of length ¢ — 1. (2) m = 3c — 1 Two chains of length ¢ and one chain of length c — 1. (3) m = 3c Three chains of length c.

If ay = ay3 = Ay = 0, a5 + O, there are four cases: (1) m = 4c — 3 One chain of length c and three chains of length c — 1. (2) nm, = 4c — 2 Two chains of length c and two chains of length ¢ — 1. (3) m = 4c — 1 Three chains of length c and one chain of length c — 1.

(4) nm, = 4c Four chains of length c.

If ay. = Qy3 = +++ = A); = 0, a), 54. # 0,7 < m, there are j cases:

(1) m =je-—j+1l One chain of length c and 7 — 1 chains of length

e—1.

(2) m =je-—-j+2 Two chains of length c and 7 — 2 chains of length e-—1.

(j) m = je j chains of length c.

When c = 2, the chains of length 2, listed above, appear.

The G»’s of class two and central of order p"* generated by H and two com- mutative operators U, and U: of order p in J have been determined.” We proceed to consider the Ge’s such that the greatest partition to which any operator in {U,, Us} corresponds is3 + 2+1+1+1+4.---. Obviously they are of class 3 and have a central of order p"-*. From Theorem 1 we see there is always a subgroup of order p in {U,, U2} of type Il, or less and we shall select one of these operators of lesser type for Uz. The commutator subgroup AK

15 H. R. Brahana, On metabelian groups, loc. cit. (footnote 11), section 6.

34 HENRIETTA TERRY

may be of order p*, p*, p*. Each case is examined for the number and type of subgroups of lesser partition in {U,, Uz}. Obviously these subgroups differen- tiate between groups which are not simply isomorphic.

However, in the groups G, with K of order p* there are 2 groups having no operators of type I and only 1 subgroup of order p of type II; in {U,, U2} which are not simply isomorphic. In one of these the commutator subgroup of {Ue2, S,} is identical with the group generated by the commutator of a com- mutator in G = {H, U,} which is a characteristic subgroup in G, and this relation does not exist in the other, hence these two groups are not simply isomorphic.

We meet this situation again in the case K is of order p* and U2 generates a subgroup of order p of type I., the only subgroup corresponding to a partition less than3 + 2+1+14414 --- in {U;, U2}.

The groups for G, of order p"*? and class 3 with a central of order p"~* con- taining H as a maximal abelian invariant subgroup of order p" extended by 2 commutative operators of order p in J are as follows:

Order of K No. of groups No. of subgroups of type

I; I,

p° 1 0 0

p* 1 1 0

1 0 1

2 0 0

p*® 1 1 1

1 1 0

2 0 1

1 0 0.

In addition to these subgroups just mentioned we have determined all the sub- groups of order p"** in the holomorph of H whose operators are all of order p, whose central is of order p"™"*", whose class is n,, whose commutator subgroup is of order p"', whose cross-cut of the central and the commutator subgroup is of order p and which contain subgroups of order p"*! of classes n; and 2 only, the commutator subgroup of the latter being of order p? at most.

The results obtained indicate the large number and variety of subgroups of order p? in J, as well as the number of abstractly distinct groups of order p"*? which contain the abelian group of order p" and type 1, 1, 1, --- invariantly. The complete characterization of the groups of order p? in J is a long and difficult task but the methods which we have developed and applied in this paper are apparently sufficient to complete it.

UNIVERSITY OF ILLINOIS.

16 The groups on lines 4 and 7 of the table are differentiated by the condition that in one of them the commutator subgroup of | U2, S,} is identical with the subgroup generated by the commutator of a commutator in G = {H, U,} which is a characteristic subgroup in G.

GENERALIZED PERFECT SETS By G. T. WuyYBURN

1. In an earlier paper' the author has defined generalized derived aggregates or K-derivatives for subsets A of a metric space C with respect to an arbitrary class K of closed sets in C. Under this definition, by the K-derivative K(A) of A is meant the set of all points z such that every neighborhood of zx contains a subset of A which is not contained in any K-set. The operation of taking K-de- rivatives may be iterated, giving successive K-derivatives denoted as follows: A = Ay, K(A) = Al, --- , K(Az7') = Af, --- , and ingeneral A% = K(A%™"')

or = |] K(A®) according as a is an isolated or a limit ordinal. B<a

This suggests the following extension of the notion of a perfect set, to which the present paper is devoted. A set of points A will be said to be K-perfect provided it is equal to its own K-derivative, i.e., K(A) = A. It results at once that, for all classes K, every K-perfect set is closed. In case K is the class of all single points, then the K-perfect sets reduce to the ordinary perfect sets.

Other examples are: (1) the Sierpinski triangle curve? is perfect with respect to the class of all simple closed curves, the class of all simple ares, or the class of all dendrites; but it is not perfect relative to the class of all regular curves; (2) the set E consisting of the curve y = sin 1/x together with the interval J = (—1,1) of the y-axis is perfect with respect to the class of all simple closed curves in E, (a vacuous class); but relative to the class of simple ares in E, this set is not perfect, since its first are-derivative reduces to J and its second are-derivative vanishes; (3) in the set F consisting of the curve y° = 2? sin? 1/x together with the origin Q, the point Q itself is perfect with respect to the class K of all simple closed curves contained in F; in fact, in this case Fi = Q, F2 = K(Q) = Q, and so on.

In what follows we shall suppose our space to be separable and metric, and K will denote an arbitrary class of closed sets in this space.

2. THEorREM. Any closed set A is the sum of a K-perfect set and a countable number of sets each of which is the intersection of A with a K-set.

Proof. Let B be the first ordinal such that A? = AfZ*'. Since the space is separable, 8 is of the first or second class. Then A® is K-perfect and we have . 8 a+l (i) A = Ax + ps [Ax — Ag’).

0sSea<38

Now for any a, each point x of AX — A*' is contained in some neighborhood Q« such that AX-Q* is contained in some K-set. Whence, by the Lindeléf

Received February 13, 1935. 1 See American Journal of Mathematics, vol. 54 (1932), pp. 169-175. 2 See Comptes Rendus, vol. 162, p. 629.

35

36 G. T. WHYBURN

theorem, for each a the set Ag — A%*' can be covered by a countable collection of neighborhoods [Q*] such that for each 7, Ag-Q* is contained in some K-set K¢. Thus

(ii) D> 4z-A4so DY YK, O0sa<gs O0Sa<8 i=l and since @ is of the first or second class, the number of sets [K‘] on the right is countable. Now (i) and (ii) together give

(iii) A=Azr+ | DA-Ki, Osa<s i=1 which proves our theorem.

Notes. (1) In case K is the class of single points, the theorem just established reduces to the classical result that any closed set is the sum of a perfect set and a countable set.

(2) If A is the whole space, or if all the K-sets are contained in A, or if every closed subset of a K-set is likewise a K-set, then all sets of the form A-K are K-sets and our theorem takes the following form: The set A is the sum of a K-perfect set and a countable number of K-sets.

(3) Sinee by definition every K-set must be nowhere dense in any given K-perfect set, and since by a well known theorem no self-compact set is the sum of a countable number of nowhere dense sets, it follows that no compact K-perfect set is contained in the sum of a countable number of K-sets. This result together with our theorem above gives the following proposition:

In order that a self-compact set A should be contained in the sum of a countable

number of K-sets, it is necessary and sufficient that A contain no K-perfect subset. . 3. It is possible for two quite distinct classes of sets K to yield the same K-perfect sets. For example, the class of all dendrites and the class of all sets which locally are dendrites in a given space yield identical K-perfect sets. Also, in general, if a class K, is contained in a class Ke, then any K»-perfect set must be K,-perfect. Similarly, any set which is perfect relative to the class of all ares in a given space must also be perfect with respect to the class of all simple closed curves in this space. In studying relationships of this sort we are led to the following

THEOREM. Given two classes K, and Kz of closed sets. If no K,-set contains a K.-perfect set, then every K2-perfect set is likewise K,-perfect.

For let A be any K2-perfect set. Then if A were not K,-perfect, there would exist a point z of A and a neighborhood U of x such that A-U is contained in

some K,-set B. But this is impossible, since B would contain A-U and it is

easily seen that A-U is K2-perfect. Notes. (1) In case the space is compact, the theorem just given takes the following equivalent form: If each K,-set is contained in the sum of a countable

GENERALIZED PERFECT SETS 37

number of K2-sets, then every K-perfect set is also K,-perfect. Thus in particular the class of all ares and the class of all are-sums (i.e., continua which are the sum of a countable number of ares) in a compact space would define the same K-perfect sets.

(2) We have seen from example (3) in $1 that for some classes K, the K-perfect sets may not be perfect in the ordinary sense. However, the theorem of this section readily yields conditions under which this is the case. For, taking K, as the class of all single points in a space C and K;, as an arbitrary class K of clused sets, our theorem tells us that in order for every K-perfect set to be perfect in the ordinary sense it suffices that every point of the set be contained in some K-set.

4. In a recent article Hurewicz has suggested the following method of taking generalized derived aggregates of a metric space X. Let f(x) be a single valued transformation defined on X and mapping X intoaspace Y. Then fis said to be stationary at the point z with respect to a subset M of X provided that all points of M in some neighborhood of x map into a single point under f. Denote by Xi(f) the set of all points of X where f is not stationary relative to X, by X2(f) the set of all points of X,(f) where f is not stationary relative to X,(f), and so on with the usual iteration and product taking. In this way we obtain a gen- eralized a“ derivative X(f) of X with respect to f.

In this concluding section it will be shown that ‘ese generalized derivatives proposed by Hurewicz are equivalent to the K-derivatives previously treated by the author in the article referred to above in $1 in the sense stated in the following theorem.

THeoreM. For any class K of closed sets in a metric space X there exists a transformation f(x) defined on X such that X.(f) = X& for all a; and conversely, for every transformation f(x) defined on X there exists a class K of closed sets such that X.(f) = Xx forall a.

Proof. To prove the first part we have only to define f(z) on X as follows:

f(x) =a, for xe(X_~ — Xg**); f(z) =B, for zxeX§, where 8 is the first ordinal such that X2 = X8*!. Then since X = Xp + (Xx — Xx) +--+ + (XE - XE") + + (a <8),

and no two of the sets on the right can intersect, f(x) is defined and single valued on X and maps X into the set of all ordinal numbers a, (0S a S 8). Nowifa is any isolated ordinal, then since f(z) = a — lon Xg~' — Xx and f(x) = aon Xx, it follows that relative to X¢-", f is stationary on exactly the set Xg-' — Xf and hence is nowhere stationary on exactly the set Xz. This gives XF = X.(f)

3 See Fundamenta Mathematicae, vol. 23 (1934), footnote to p. 54.

38 G. T. WHYBURN

for all such a’s, and a simple application of transfinite induction yields this same relation for all ordinals a.

Now conversely, if we are given f(x) defined on X, then if we let K be the class of sets [f—'(y)] for all yef(X), we will have X.(f) = XX for alla’s. For ifa is any isolated ordinal, and z is any point of X.-:(f) where f is stationary relative to X.-:(f), there exists a neighborhood V of X such that V-X,_:(f) is contained in a K-set f-'(y), whereas if f is not stationary at z relative to X._,(f), then no such neighborhood of z exists. This gives Xk = X.(f) for all isolated a’s, and again the principle of transfinite induction yields this relation for all a’s.

Tue UNIVERSITY OF VIRGINIA.

RIEMANNIAN MANIFOLDS IN THE LARGE By SuMNER Byron Myers

1. Introduction. The general problem to be considered here is that of determining relations between the local properties of an analytic Riemannian manifold and the topological properties of the manifold in the large. In par- ticular, we are interested in determining topological properties from a knowledge of local properties in the neighborhood of just one point, and conversely, in deter- mining possibilities of metrisation of a given topological manifold by means of local analytic Riemannian geometries.

By an n-dimensional analytic Riemannian manifold in the large will be meant a complete manifold in a sense to be defined later, equivalent to the “normal” Riemannian space of Cartan! and a generalization to n dimensions of the ‘“‘com- plete surface’ of Hopf and Rinow.?

The results obtained here are in most cases generalizations of theorems given in the two-dimensional case by Hopf and Rinow. A complete summary of known results on the general problems stated above, as well as a statement of some specific unsolved problems can be found in a paper by Hopf entitled Differentialgeometrie und topologische Gestalt, Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 41 (1932), pp. 209-229. Some of the problems proposed there (p. 222, lines 14-22, and p. 224, lines 25-29) are solved in the present paper.

In §2 we define complete manifolds. §3 contains a proof that a complete manifold whose Riemannian curvature at every point and with respect to every plane direction is greater than a positive constant e is compact and has a diam- eter less than z/e!. In §$4 is proved the fundamental uniqueness theorem—a given n-dimensional Riemannian element E can be continued to (i.e., contained in) at most one complete, simply connected n-dimensional manifold. In §5 we set up certain codérdinate systems in FE and give necessary analyticity condi- tions that E may be continued to a complete manifold. We also show how to determine from a certain coédrdinate system in the element E about a point A the points conjugate to A. In §6 we show that under certain analyticity conditions an element EZ about a point A without conjugate point can be continued to a complete manifold homeomorphic to n-dimensional space S,, from which fol- lows that if a complete simply connected n-dimensional manifold contains a point without conjugate point, it is homeomorphic to S,. Finally in §7 we

Received by the Editors of the Annals of Mathematics, February 27, 1934, accepted by them, and later transferred to this journal; presented to the American Mathematical Society December, 1933.

1E. Cartan, Lecons sur la géométrie des espaces de Riemann, Paris, 1928.

2H. Hopf and W. Rinow, Uter den Begriff der vollstdéndigen differentialgeometrischen Fldache, Commentarii Mathematici Helvetici, vol. 3 (1931).

39

40 SUMNER BYRON MYERS

prove that if the universal covering manifold of a complete manifold is not compact, through every point A passes at least one geodesic ray without con- jugate point to A.

2. Complete analytic n-dimensional Riemannian manifolds.’ We are con- cerned here with topological n-dimensional manifolds (n = 2). These are connected separable Hausdorff spaces, in which each neighborhood of the de- numerable set of neighborhoods used to define the space is homeomorphic to the interior of the (n — 1)-dimensional unit sphere. Only intrinsic properties of these manifolds will be used, so that no imbedding in euclidean spaces is neces- sary. The words closed and open as applied to these manifolds are synonymous with compact and not compact in the usual sense.

If each of the neighborhoods is provided with a coérdinate system so that in the region common to two intersecting neighborhoods one set of codrdinates can be obtained from the other by means of an analytic transformation with non-vanishing jacobian, the manifold is called analytic. Any coérdinate system which is obtainable from one of the above by means of an analytic transforma- tion with non-vanishing jacobian is said to be admissible.

Now suppose that the manifold is metrisable in the following manner. To each neighborhood with admissible coérdinate system (x) = (x, --+ ,2%,), we assign a real analytic positive definite symmetric quadratic differential form

Jas dXqdx3 (a, B = 1, ee n).

The functions appearing in these quadratic forms are to have the property that if (x) and (#) are admissible coérdinate systems respectively in two intersecting neighborhoods, and gas(x)dradzg and Gas(%)dé.dé3 are the quadratic forms re- spectively assigned to them, then in the intersection of the two neighborhoods,

Gas(x)dtadxg = Gas(t)diadks

under the transformation between (x) and (%). We define are length along a curve as the integral of the square root of the quadratic differential form. We now have an analytic n-dimensional Riemannian manifold M.

If we define the distance between two points as the lower limit of the lengths of rectifiable curves joining the two points, the manifold M satisfies the axioms for a metric space. Furthermore, the idea of neighborhood resulting from this notion of distance coincides with the original topological notion of neighborhood.

In treating such manifolds in the large we restrict ourselves to manifolds which are complete. Our definition of completeness is identical with that given by Hopf and Rinow in the two-dimensional case, and is as follows.

An analytic n-dimensional Riemannian manifold M is said to be complete if it satisfies any one of the following four equivalent postulates:

(1) Every geodesic ray can be continued to infinite length on M.

3 Statements made in §2, requiring proof, for which no proof is given, are formal gener- alizations of corresponding 2-dimensional theorems and proofs of Hopf and Rinow, loc. cit.

RIEMANNIAN MANIFOLDS IN THE LARGE 41

(2) Every divergent* line is infinitely long.

(3) Every fundamental sequence’ of points on M converges.

(4) Every bounded set of points in M has a limit point in M.

Postulates (1) and (2) have been used in the treatment of analytic Riemannian manifolds of constant curvature. Postulate (3) makes M a complete metric space in the sense of Hausdorff,’ while postulate (4) (the Weierstrass-Bolzano theorem) is more restrictive than (3) for general metric spaces. However, for analytic Riemannian manifolds the four postulates are equivalent.*

Throughout the rest of this paper the word manifold will be used to denote an analytic Riemannian manifold. j

An n-dimensional manifold M which is complete in the sense just defined is an entire or non-continuable manifold. By this we mean that M is not a proper subset of another n-dimensional manifold M’. However, the class of complete manifolds is smaller than the class of non-continuable manifolds. An example of an incomplete but non-continuable n-dimensional manifold is the universal covering manifold M of the manifold M obtained by removing an (n — 2)- dimensional euclidean space from n-dimensional euclidean space. It is under- stood that the neighborhood of each point of M is to be provided with the euclidean geometry of the neighborhood of M which it covers. Similar examples are the universal covering manifolds respectively of the manifold obtained by removing an (n — 2)-dimensional hyperbolic space from n-dimensional hyper- bolic space, and of the manifold obtained by removing an (n — 2)-sphere from the n-sphere.

Nevertheless, the class of complete manifolds is not too small for our purposes. It contains all closed manifolds, and also all open manifolds which can be regu- larly imbedded in a euclidean space and are closed in that space. Furthermore, a number of fundamental theorems can be proved for complete manifolds but not for the mere general class of non-continuable manifolds.

A first such theorem is the following:

THEOREM 1. Every two points on a complete manifold can be joined by a curve of shortest length, and this curve is a geodesic.

The examples given above illustrate the fact that this theorem does not hold if the hypothesis of completeness is replaced by that of non-con- tinuability.

* By a divergent line on a manifold M is meant the single-valued continuous image of a straight line ray if to every divergent sequence of points on the ray corresponds a divergent sequence of points on M.

5 A fundamental sequence of points x; in a metric space is a sequence for which the dis- tance p(zj, 2;4:) satisfies the Cauchy criterion.

® See H. Hopf, Zum Clifford-Kleinschen Raumproblem, Math. Annalen, vol. 95 (1926), pp. 313-315.

Also see P. Koebe, Riemannsche Mannigfaltigkeiten und nichteuklidische Raumformen, Sitzungsber. Preuss. Akad. d. Wiss., Phys.-math. Klasse, Berlin, (1927), pp. 184-185 and (1928), pp. 349-350.

7 See F. Hausdorff, Mengenlehre, p. 315.

’ The proof of this fact, and the proof of Theorem 1 which follows later, are formal

42 SUMNER BYRON MYERS

3. Relations between curvature and topological properties. The following are well known results on space-forms (manifolds of constant curvature).

(A) For every n and every K there exists a unique? complete simply connected n-dimensional space-form with curvature K. According as K > 0, = 0, or < 0 this manifold is a spherical, euclidean, or hyperbolic space.

(B) A complete n-dimensional space-form with positive curvature is closed.”

We give now a statement and proof of a theorem which is a generalization of (B) to manifolds of non-constant curvature and at the same time a generalization to n dimensions of a 2-dimensional theorem of Hopf and Rinow (loc. cit., p. 224, Satz V). This result includes a generalization of the classical Bonnet theorem on the diameter of an ovaloid."

THEOREM 2. A complete n-dimensional manifold M whose curvature is greater than a positive constant e at every point and with respect to every plane direction is closed and has a diameter less than x/e}.

Consider any geodesic are g on M. We can set up codrdinates (7) = (x,

- , ,) in the neighborhood of g such that the functions gas(z) (a, B = 1, --- , ) appearing in the fundamental quadratic form satisfy the follow- ing conditions along g:

Jas = Sag

(3.1) Ages _ ‘ (a, B,y = 1,---,n) Ory

while the coérdinates 2, --- , ,-1 are constant along g and z, is the are length

s measured from any point A on g.” The points on g conjugate to A are given by the zeros of a determinant whose

columns are n — 1 linearly independent sets of solutions m(s), --- , m,-:(s) of the equations

(3.2) 1; + Rnurni(s)m = 0 (G,k =1,---,—1) vanishing at s = 0 but not identically zero. The functions Rag,,s(s) (a, 8, y, 6 = 1, --- , m) are Riemann symbols of the first kind in the coérdinates (x) taken along g.

generalizations of the 2-dimensional proofs given by Hopf and Rinow, loc. cit., pp. 215-221. Fundamental in the proof is the fact that every point of a Riemannian manifold has a neighborhood in which every two points can be joined by a unique geodesic arc of shortest length, which has been proved for n dimensions by J. H. C. Whitehead, Quarterly Journal of Mathematics, vol. 3 (1932), pp. 33-42.

® That is, unique except for isometric manifolds.

10 See H. Hopf, Math. Annalen, vol. 95 (1926), pp. 313-339.

1 A generalization of the Bonnet theorem to n dimensions has previously been given by I. J. Schoenberg, Annals of Math., vol. 33 (1932), pp. 485-495. In the present paper the hypothesis of completeness enables us to draw the topological conclusion that the mani- fold is closed.

2 See T. Levi-Civita, Math. Annalen, vol. 97 (1927), pp. 291-320.

RIEMANNIAN MANIFOLDS IN THE LARGE 43

The curvature of M at a point P on g with respect to the plane direction defined by the unit vector (0, 0, --- , 1) tangent to g at P and any unit vector

(uw, +--+ , Una, 0) orthogonal to g at P is given by

(3.3) Rak nit ite (i,k =1,---,n— 1). Then according to hypothesis, along g

(3 .4) Rix nitite > € Gi,k =1,---,n —1) for all numbers (uw, --- , U,—1) satisfying the relation

(3.5) = uz =1.

We may write the inequality (3.4) as follows:

n—l

(3.6) Rurnitite > >, u? (j,k =1,---,n—1).

i=l

Let us compare the two sets of equations

(3.7) n, a Rak nink = 0 (i, k= 1, + 8 = 1) and (3 .8) n, + en: = 0 @=1,---,n— 1).

According to the generalization of the Sturm comparison theorem given by Morse," we can deduce from (3.6) that the first conjugate point of A on g is at a distance less than +/e} from A. Thus any geodesic are on M of length x/e! or greater contains a point conjugate to its initial point, and is not the shortest are joining its end points.

Hence the geodesic of shortest length which, according to Theorem 1, exists between any two points of the complete manifold M, is shorter than x/e!. The diameter of M is thus less than z/e!, and M is a bounded manifold. A use of the fourth completeness postulate enables us to conclude that the boundedness of M implies its compactness, that is, M is closed.

4. The uniqueness theorem. By an n-dimensional Riemannian element FE we shall mean a point and its neighborhood homeomorphic to the interior of the (n — 1)-dimensional sphere, and provided with an analytic Riemannian geom- etry, that is, with a coérdinate system and an analytic, positive definite, sym- metric Riemannian quadratic form. We are concerned with determining the topological properties of any complete manifold to which a given element may be continued.

Among all the manifolds the simply connected ones play an important role. For if we have an arbitrary manifold M containing an element Z£, we can obtain

13 Morse, Math. Annalen, vol. 103 (1930), p. 66.

44 SUMNER BYRON MYERS

a simply connected manifold also containing the element EF by providing each neighborhood of the universal covering manifold M of M with the Riemannian geometry of the neighborhood of M which it covers. If M is complete, M will also be complete.

The following theorem, a generalization of Theorem (A) on manifolds of constant curvature, is fundamental.“

THEOREM 3. Every n-dimensional Riemannian element E can be continued to at most one complete, simply connected n-dimensional manifold, that is, if two such continuations of E exist, they are isometric.

Suppose that we have two complete simply connected continuations M and M’ of the element E. Then the neighborhood N of a certain point A on M will be isometric to the neighborhood N’ of a point A’ on M’. Any admissible coordinate system in N and its quadratic differential form can be used in N’ by giving points in N’ the same coérdinates as their correspondents in N.

Let g be a geodesic are issuing from A, and g’ the corresponding geodesic arc issuingfrom A’. We can set up the coérdinates (x) of §3 in the neighborhood of gon M. In the neighborhood of g’ on M’ we can set up a similar coérdinate system (x), with the property that (x’) = (x) for points in N and N’ correspond- ing under the given isometry.

We will use R,s.ys(2) to denote the Riemann symbols of the first kind in the coérdinates (x). Then Ry »i(O0, --- , 0, shu, (, k = 1,---,n — 1) is the curvature of M with respect to the plane direction defined by the unit vector tangent to g at P and the unit vector (wm, --- , u,+, 0) orthogonal to g at P; that is, the curvature at P of the geodesic surface S determined by the direction of g at P and the direction (uw, --- , uns, 0). Now in the coérdinates (xr) the Christoffel symbols are all zero along g, and hence if we move the vector (uw, «++ » Una, 0) along g parallel to itself with respect to g and M, the com- ponents uw, --- , U,—1, 0 remain constant. But according to the theorem of Severi we get the same result if we move the vector (w, --- , U,—1, 0) along g parallel to itself with respect to g and S. Thus the same geodesic surface S is determined by the direction of g at each of its points and the direction (us, --+ , Un—1, 0). Hence Ryxniuiue is the curvature of the surface S along g.

We conclude that R,x.:(s)uiu, is an analytic function of s for all values of s > 0" and for all (uw, ---,u,-1). Therefore R,x.»: is an analytic function of s for all positive values of s. Similarly, on the manifold M’ the Riemann sym- bols R;,.,; in the codrdinates (x’) are analytic functions of s along g’.

But along the ares of g and g’ lying in N and N’ respectively Rain: = Rii.ni- Hence Rnini = Rix»; for all values of s > 0.

Now the functions Ry »:(s) and R{,.,;(s) determine the points on g and g’

144 For n = 2, this theorem and the following theorems have been proved by W. Rinow, Uber Zusammenhange zwischen der Differentialgeometrie im Grossen und im Kleinen, Math. Zeit., vol. 35 (1932), pp. 512-528. J. H. C. Whitehead has given a generalization of (A) in a different direction to locally homogeneous spaces with a Lie pseudo-group. See Annals of Math., vol. 33 (1932), pp. 681-687.

18 Since M is complete, the geodesic are g can be extended to infinite length.

RIEMANNIAN MANIFOLDS IN THE LARGE 45

conjugate to A and A’ respectively, through equations (3.2). Hence the points on g conjugate to A are at the same distance from A as the points on g’ conjugate to A’ are from A’.

Let AB be a smooth segment of g; that is, a segment without multiple points and without a point conjugate to A. The corresponding segment A’B’ of g’ will, by the result of the preceding paragraph, contain no point conjugate to A’. Then there exists an (n — 1)-parameter family of geodesics on M through A

(4.1) La = La(Yi, *** 9 Yn-1, 8)

containing g for (yi, --- , Yaa) = (0, --- , 0) and forming a field F” in the neighborhood of AB. If A’B’ has no multiple points, it can be imbedded in a similar field F’ by means of the corresponding field of geodesics through A’. The parameters y;, --- , y,_; in this latter family can be taken so that geodesics through A and A’ corresponding under the given isometry between the neighbor- hoods N and N’ of A and A’ respectively are determined by equal parameters (yi, aig Yn—1) = (v1, iets Yn—1).

In the field F we can use as coérdinates of a point P n numbers (yw, --- , Yn) the first » — 1 of which are equal to the parameters in (4.1) determining the geodesic AP of the family (4.1) and the last of which is equa! to the length of the geodesic are AP. In F’ we can choose similar coérdinates (y;, --- , y,). Points in F-N and F’-N’ which correspond under the given isometry will have equal coérdinates, and hence the Riemannian quadratic forms in the variables (y) and (y’) will be the same in F-N and F’-N’, therefore the same throughout F and F’. If we make points in F and F’ with the same coérdinates correspond, the correspondence will be an isometry between F and F’, and will be a continuation of the given isometry between N and N’.

If A’B’ has multiple points, the family of geodesics through A’ forms a field im kleinen. The variables (y’) can be used as coérdinates im kleinen, and by making points with equal coérdinates correspond, we obtain a single-valued, im kleinen isometric map of the field F on the multiple-leaved field around A’B’.

Thus the isometric map of N and N’ can be continued along any smooth geo- desic arc issuing from A. Furthermore, if P is any point on M which can be reached in this manner, the isometry can be continued along any smooth geodesic arc issuing from P. Since any geodesic are, or any broken geodesic arc, is com- posed of a finite number of smooth geodesic ares, the isometry can be continued from A along any such are.

Any point Q on M can be connected to A by a geodesic are. Hence the neighborhood of any point Q of M can be mapped isometrically on a neighbor- hood of a point Q’ of M’.

We must now show how to continue the isometry along an arbitrary are AP issuing from A. This is done as in the 2-dimensional case” by constructing a closely approximating broken geodesic with corners on AP.

16 An improper field. The field breaks down at A. 17 See Rinow, loc. cit., pp. 516-518.

46 SUMNER BYRON MYERS

Finally, an application of the Monodromiesatz shows that the im kleinen isometric map of M on M’ is single-valued, and since the réles of M and M’ can be reversed, the map is one-to-one. Thus we have an isometry between M and M’.

Corouuary 1. Let M’ be a complete n-dimensional manifold, and M an ar- bitrary simply-connected n-dimensional manifold. If M’ and M are continuations of the same Riemannian element E, then M can be mapped in a single-valued and im kleinen isometric manner on a portion of M’.

The proof is included in the proof of the theorem.

Coro.tuary 2. If two complete n-dimensional manifolds are continuations of the same Riemannian element E, then their universal covering manifolds are homeomor phic.

5. Conditions that an element can be continued to a complete manifold. Not every analytic Riemannian element can be continued to a complete manifold. Some necessary conditions are given in the following theorems.

THEOREM 4. (Given an n-dimensional Riemannian element E around a point A. Sei up the geodesic codrdinates (x) of §3 along any geodesic arc issuing from A. Then if E can be continued to a complete n-dimensional manifold, the Riemann symbols Ryxni in the coérdinates (x) taken along g must be analytic functions of the arc length x, = s for all positive values of s.

This has already been shown in the proof of Theorem 4.

THEOREM 5. Given an n-dimensional Riemannian element E around a point A. Set up the codrdinates (y) of § 4 along the neighborhood of any geudesic are g issuing from A, thus obtaining a fundamental quadratic form of the type dasdyadys (a, 8 = 1, --- ,n) in whicha»w = law = O0(¢ = 1,---,n —1). Thenif E can be continued to a complete n-dimensional manifold, each function ai; (i, j=1,--+,n — 1) takenalong g must be an analytic function of the arc length Yn = 8 for all positive values of s.

The transformation from the coérdinate system (x) in the neighborhood of g to the coédrdinate system (y) is given by the equations (4.1) of the (n — 1)- parameter family of geodesics through A. Each column of the determinant | dx;./dy; | taken along g represents a set of variations of the family of geodesics (4.1) along g, and hence is a solution of equations (3.2), since the latter are the equations of variation of the geodesics.

According to Theorem 4, R,x,.; is an analytie function of s for all s > 0. Hence solutions of (3.2) are analytic functions of s for alls > 0. In particular, along g d2,/éay; (j,k = 1, --- , m — 1) is an analytic function of s for all s > 0.

But along g

nie OX; OX,

(5.15) hak (i,j,k =1,---,n—1).

Therefore along g a;; is an analytic function of s for alls > 0. This completes the proof.

RIEMANNIAN MANIFOLDS IN THE LARGE 47

As in the case of polar coérdinates on two-dimensional manifolds," it can be shown that the functions an, --- , @—1,.-1 vanish at A. It follows from (5.15) that 6z,/dy; (i, k = 1, --- ,m — 1) vanish at A. Hence | dz;/dy; | is a deter- minant whose zeros s ¥ 0 define the points on g conjugate to A. But by means of (5.15) we see that along g

| Aa. s

| OY;

We have, then, the following theorem:

THEOREM 6. The points on g conjugate to A are given by the zeros s # 0 of the determinant | aj; | taken along g.

THEOREM 7. (Given an n-dimensional element E around a point A. Set up coérdinates (y) as in Theorem 5 along an arbitrary geodesic g through A. Then if E can be continued to a complete n-dimensional manifold M, a,; will be an ana- lytic function of the n variables (y:, --- , Yn) for 0 < yn < K, where K is the first value of yn for which | aj; | vanishes on g, and for (yi, «++ , Yn—1) in the neighborhood of their values on g.

According to Theorem 6, y, = K gives the first point on g conjugate to A. Hence the geodesics through A form a field in M in the neighborhood of g for 0 < y, < K. We can use the coérdinates (y) throughout this field, and the functions a;; will be analytic throughout the field.

Corotiary. If | a,; | has no zeros on g, then if E can be continued to a com- plete manifold, a;; is an analytic function of (y:, --+ , Yn—1) in the neighborhood of gfor0 < yn < @,

(5.16) |a;| =

6. Conjugate points and manifolds in the large. The following theorems give connections between the existence of points conjugate to a point A and the topological properties of the n-dimensional manifolds to which the n-dimensional element around A can be continued.

THEOREM 8. Given an n-dimensional element E around a point A. Suppose that the coefficients a;; in the Riemannian quadratic form for coérdinates (y) along the neighborhood of an arbitrary geodesic g issuing from A are analytic functions of (y) for (y1, +++ 5 Yn—1) in the neighborhood of their values on a and for 0 < y, < 2. Then if A has no conjugate point, the element E can be continued to a complete n-dimensional manifold homeomorphic to n-dimensional space.

Since A has no conjugate point, according to Theorem 6 the determinant | a,; | is different from zero for (y, --+ , yn) near their values on g and for 0 < y. < , and hence the quadratic form ajdydy; + dy? will be positive definite throughout this region of the variables (y).

These coérdinate systems along the varicus geodesics through A overlap, and in a region of overlapping in E one coérdinate system can be obtained from the other by an admissible transformation of coérdinates keeping invariant the

18 See Blaschke, Vorlesungen tiber Differentialgeometrie, vol. I, 1930, p. 152.

48 SUMNER BYRON MYERS

Riemannian quadratic form, since all these codrdinate systems were got directly by admissible transformations from one original coérdinate system in the element £.

Now the geodesics through A in the element E can be put into a one-to-one correspondence with the straight lines through a point in n-space. The initial segment of each such straight line ray and its neighborhood can be provided with the Riemannian geometry belonging to the corresponding geodesic through A and its neighborhood in E. By the remarks in the first paragraph of this proof, this Riemannian geometry can be extended all along the neighborhood of the straight line. The relation between the codrdinate systems of two such overlapping neighborhoods is determined by the corresponding overlapping in EZ.

The result is an n-dimensional analytic Riemannian manifold M containing the element Z, and homeomorphic to n-space. The manifold M is complete; for every bounded set of points lies within the interior of a geodesic hypersphere of finite radius about A, and hence has a limit point on M.

TuHeoreM 9. If a complete simply-connected n-dimensional manifold contains a point without conjugate point, then it is homeomorphic to n-dimensional space.

Let the manifold be called M and the point without conjugate point be called A. Then by Theorem 6 the determinant | a;; | formed for coérdinates (y) in the neighborhood of an arbitrary geodesic are issuing from A does not vanish. Hence, by the corollary to Theorem 7, the hypotheses of Theorem 8 are satisfied, and the element of M around A can be continued to a complete n-dimensional manifold homeomorphic to n-dimensional space. From the uniqueness theorem (Theorem 3) we deduce that M is homeomorphic to n-dimensional space.

Corouuary 1. If a complete n-dimensional manifold M contains a point A without conjugate point, then the universal covering manifold of M is homeomorphic to n-dimensional space.

For the element around A can be continued to the universal covering mani- fold of M, which will be complete. An application of Theorem 10 proves the corollary.

CoroLuary 2. Through every point A of a complete n-dimensional manifold whose universal covering manifold is not homeomorphic to n-dimensional space passes a geodesic containing a point conjugate to A. In particular, through every point A of a simply-connected, closed, n-dimensional manifold passes a geodesic containing a point conjugate to A.

THEOREM 10. Through every point A of a complete open n-dimensional manifold M passes a geodesic which is the shortest line between A and any of its points, and hence contains no point conjugate to A. Through every point A of a closed n- dimensional manifold M whose universal covering manifold is open passes a geo- desic containing no point conjugate to A.

If M is open, it will contain a sequence of points P; without a limit point. Let A be an arbitrary point on M. Since M is complete, the distance p(A, P;) from A to P; becomes infinite as 7 becomes infinite. Furthermore, we can pass a geodesic c; of shortest length between A and P;. Let p(A, Pi) = ri.

RIEMANNIAN MANIFOLDS IN THE LARGE 49

Interior to a neighborhood of A simply covered by the geodesics through A, consider a geodesic hypersphere H. Corresponding to the sequence of points P; and geodesics c; there will be a sequence of points P’, on H at which the geo- desis ¢; intersect H. The sequence P; will have a limit point P’on H. Let c be the geodesic AP’, Q any point on c, and L the length of AQ onc. A certain subsequence P{ of P; will converge to P’. Let é; be the geodesic joining A to P:, and P; be the subsequence of P; on the geodesics @;. Then for sufficiently large i the lengths AP; = 7; on the geodesics @; will be greater than L.

Measure off the length L on the geodesics @; thus getting points Q;. Since é; is a geodesic of shortest length between A and P,, and since 7; > L, é is also a shortest line from A to Q,. Hence p(AQ,) = L.

The points Q; converge to the point Q, because of the continuous dependence of the geodesics through A on the points of H through which they pass. There- fore p(A, Qi) — p(A, Q), and hence p(A, Q) = L. But L was the length of AQ on ¢, so that we have proved that c is the shortest line from A to Q.

But Q was any point on c. Thus c is the shortest line from A to any of its points, and contains no point conjugate to A.

The second part of the theorem is proved as follows. If M is a closed n- dimensional manifold whose universal covering manifold M’ is open, then we can apply the first part of the theorem to M’. Hence through every point A’ of M’ passes a geodesic c’ without a point conjugate to A’. Now M’ can be mapped on M in a single-valued, im kleinen isometric manner. If A is the point of M corresponding to A’, and ¢ the geodesic corresponding to c’, then ¢ can contain no point conjugate to A. Since every point A of M corresponds to some point A’ of M’, the theorem is proved.

Corotuary 1. If a complete n-dimensional manifold M contains a point A such that every geodesic through A contains a point conjugate to A, then the uni- versal covering manifold of M is closed.

CoroLuary 2. An n-dimensional element E about a point A such that on every geodesic through A there exists a point conjugate to A cannot be continued to a com- plete open n-dimensional manifold.

HarvarD UNIVERSITY.

ON THE HOMOLOGY CHARACTERS OF SYMMETRIC PRODUCTS

By Moses RICHARDSON

In the first part of this paper, we consider a complex K, subjected to the trans- formations of a group of finite order p. We define a new complex k, the so- called “domain of discontinuity” of the group, by identifying all points which are images of each other under the transformations of the group. We then determine the Betti numbers of k, both non-modular and mod zr“, 7 a prime and not a factor of p. In the second part, we use the results of Part I to obtain explicit formulae for the Betti numbers of 2-fold and 3-fold symmetric products of a complex in terms of its own Betti numbers, and we indicate a general pro- cedure by which the Betti numbers of a q-fold symmetric product can be com- puted. In the third part, we continue the development of the theory of the 2-fold symmetric product. The methods of Part II yield the Betti numbers mod zx“ when z is an odd prime but not when t = 2. The Betti numbers mod 2 have been found by P. A. Smith.'’ By an extension of his methods, we determine the Betti numbers mod 2", u > 1. With a knowledge of all the Betti numbers modulo powers of primes, the torsion coefficients can be determined.?

Part I

1. Consider a simplicial oriented n-complex K. Let G be a group, of finite order p, of (1, 1) continuous transformations of K into itself. The transforma- tions J = To, T;, --- , T»-1 of G will be subject to the following restrictions:

(a) they carry m-simplexes of K into m-simplexes of K;

(b) no Ty (A = 0, 1, --- , p — 1) earries a vertex into an adjacent* vertex. As a result of (b), a simplex can be invariant only if it is pointwise invariant.

If En = VoVi --- Vm is an oriented* m-simplex of K, we define T)E,, to be the oriented simplex T,Vo7T,Vi --- T,Vm of K. Likewise, if C is the chain StE;, we define T,C to be the chain Dt;7,E;. Since

En — 2 (— 1)° Vo pete Vou Vous ed Vin s=0

Received by the Editors of the Annals of Mathematics, May 25, 1934, accepted by them, and later transferred to this journal.

1P. A. Smith, The topology of involutions, Proc. Nat. Acad. Sci., vol. 19 (1933), pp. 612- 618. The author is indebted to Prof. Smith for much stimulating guidance received during the preparation of this paper.

2A. W. Tucker, Modular homology characters, Proc. Nat. Acad. Sci., vol. 18 (1932), pp. 467-471.

3 Two vertices are called adjacent if they are distinct and if both occur in the symbol of the same simplex.

Our notation and terminology will be largely that of S. Lefschetz, Topology, Amer. Math. Soc. Colloquium Publications, No. 12, New York, 1930.

50

HOMOLOGY CHARACTERS Ur SYMMETRIC PRODUCTS 51 and T, E.— > (— 1)" T, Vo --- Ty Vow Tr Vays +++ Ti Vn, it is clear that T,F(E,,.) = F(T\E,). Hence T,F(C) = F(T,C). 2. To each of the points V;, T:Vi, --- , T>-1Vi we shall associate the mark v; and shall write AT, V; = v; (A = 0,1, --- ,p — 1). Suppose V; and V; are

such that V; T, V; (A = 0, 1, :-- , p — 1); in particular, this condition is satisfied whenever V; and V; are adjacent, by (b), §1. Then the sets

(Vi, T.V i, is T,-Vi)

and

(Vi, TiV i, dete T,»-V;) have no common element; hence v; ¥ v;._ It follows that if Vo, Vi, --- , Vm are the vertices of a non-oriented simplex | E,, | = | VoVi--- Vm |, then vo, %1, «++ , Um

are distinct and can therefore be taken as the vertices of an m-simplex. If En is the oriented simplex VoV; --- V,., we define AZ,, to be the oriented simplex AVoAV, --- AV = Vod1 +++ Um = Em, Say. If | VigVi, --- Vi, | is a k-face of | En |, then it is clear that | v;,0;, --- v;, | is a k-face of | ém |; hence the totality of oriented simplexes e} = AE‘ (m = 0,1, --- , n; i = 1, 2, ---) constitutes an oriented n-complex which we shall denote by k. Our purpose is to determine the Betti numbers of k.

If C = dt:Ei, we define AC to be the chain c = St;AE{, = Stye'. Since

oe dX (- 1)*¥o «++ U1 Vs41 «++ Um, it is clear that F(AE,) = AF(E,).

Consequently, A preserves bounding relations. Thus, for example, if C ~ 0, then AC ~ 0. Let e,, be an oriented m-simplex of k. We have

€m = ATE», (A = 0, 1, sp 1).

p—l

We now define A’e, tobe the chain }> T,E,, on K. And if c = St,ei, then

A=0

we define A’c to be the chain Dt;A’e} on K. It is easily seen that if e,, -— Ste}, then A’e, — Zt:A’ei_,. Thus A’ also preserves bounding relations. It is to be noted that A and A’ are not the inverses of each other. We have instead the relations

p—1 (2.1) VAC = >> 7AC,

A=0 (2.2) AA’c = pe.

Note that everything said so far may be understood to be mod q without any change. For example, A and A’ preserve bounding relations mod q.

52 MOSES RICHARDSON

3. Let ™,T?, --- , I be the cycles of a minimal base with respect to homology mod x“, x a prime, of m-cycles of K. All homologies in this section are under- stood to be mod x“, and all equations are understood to be congruences mod zr“. Let ™, I, ---,I", r = R,,(K), be the elements of a minimal base for weak homology® and I''t!, ['7+*, .-- , I the elements of a base for zero-divisors. Note that I‘, 7 < r, is an element of order® x“, while I‘, i > r, is an element of

order 7°, a <u. We have

(3.1) TT. ~ > =a}, Ti (= 1,2,---,s;X =0,1,---,p—1), i=l

where the determinant of each matrix ("a} ;) is +1 and the matrix ("a ;) is in

p-l fact the identity matrix (5;;) of orders. Consider the matrix (z7;) = >> (a> ;). A=0

We may suppose that the first p rows of (x7) are linearly independent mod r“, where p = p(x”) is the maximum number of rows, or columns,’ of (x7 ;) linearly independent mod x“. Note that p < r since the zero-divisors cannot contribute independent rows.

(3.2) If (sux) = (tij)(uix), then p(s) S p(u). For, suppose that

Sity Sig) +++ y Sip iy where p = p(u), were independent columns. We can pick integers

C1, Co, hie Co +ly

etl not all congruent to zero, such that > c, uj, = 0 for all values of 7. Then k=1

etl e+) etl >» Cy Sik = > i Ck 2. ti; Uj = >» ( caus) ti; = 0 e=1 k=1 i i \k=1 for all values of 7.

We can now prove the

TuHeoreM 1. If p is not divisible by 7, then Rn(k, x“) = p(x”).

Proof. (A). There exist p m-cycles mod r“ of k, namely, the cycles y! = AT",

vy? = AT?, --- , y? = AI’, which are linearly independent with respect to ho- p mology mod x“. For, suppose that >> t,7’ ~ 0, where some ¢; ~ 0. Then i=1 p e-} > t;A’y' ~ 0, or, by (2.1), > ti D> TT! ~ 0. By (3.1) we have i=l i=l A=0 p p—li Yt "ar ~ 0, i=1 A=0

5 Following P. Alexandroff Hinfachste Grundbegriffe der Topologie, Berlin, 1932, we shall say C is ‘‘weakly homologous to zero’? when C = 0.

6 The order of a cycle C is the least positive integer ¢ such that tC ~ 0.

7 The maximum number of rows linearly independent mod x“ is equal to the maximum number of columns linearly independent mod 7“, since reduction of the matrix to canonical form preserves both numbers and is unique. Cf. J. W. Alexander, Combinatorial analysis situs, Trans. Amer. Math. Soc., vol. 28 (1926), pp. 301-329.

HOMOLOGY CHARACTERS .OF SYMMETRIC PRODUCTS 53

& p or >> t,2", I? ~ 0 which implies that >> tx}; = 0 for all values of 7. But this i=1 $=] contradicts the hypothesis that the first p rows of (27) are independent. (B). Any p + 1 arbitrary m-cycles mod x", say 6', &, --- , 6°*!, of k are

linearly dependent with respect to homology mod x“. For we can write A’d' = p—l

Di = >> 7,4‘, where A‘ is a chain such that AA‘ = 6‘. Since Dé is a cycle, A=0 we have Di ~ }> »,;T' (i = 1,2, --- ,p +1). Now, by (3.1), gi

TDi ~ Yo yi Tri ~ DY yma), I. i,k Thus, — — k i m xk m wk xk . - nD ~ DY wii a}el* = Do yjtl*® = Do zal", say. A=0 ik A=0 ik k

Since T, D' = D‘ we have pD' ~ 3 2z%.T*. Since zy. = Zz. yijx", we can find inte- p+1 : gers t, te, --- , t41, not all zero, such that p tizi, = 0 for all values of k, by 1

(3.2). Therefore p 2t;D' ~ Yt;zxT* = 0. Hence, operating with A, we have

prt:AD' ~ 0 or p> Std’ ~ 0. Let t be the g.c.d. of t), te, --- , 41. The eyeles

6', 6, --- , 6°*! are dependent provided that p?t # 0. In particular, since

t ¥ 0, this condition is satisfied if p # 0 mod x. This completes the proof. The following two corollaries are immediate.

(3.3) If pis not divisible by x, and R,,(K, x“) = 0, then R,,(k, x“) = 0.

(3.4) If p is not divisible by x, and K is an n-sphere, then

R» (k, x“) = 0 (m = 1,2,---,n— 1). (3.5) In the non-modular case we need not consider the entire base ™, r*. weey Ts

for homology of m-cycles. It will in fact be more convenient in the applications to follow in Part II of this paper to use only the cycles [, I, --- , 0", r = R,.(K), which form a minimal base for weak homology, and to employ the symbol =

throughout. Thus, we write T,I'; = >> "a},rj, (@@ = 1, 2,---, 7) and

i=l p-l1 (z7;) = > (™a*,). Then, in the non-modular case, R,(k) = p(x™) where p is A=0

the ordinary rank of (x’?;). The proof follows exactly the same lines as that of Theorem 1. The corollaries analogous to (3.3) and (3.4)° also hold in the non- modular case.

8 The corollary analogous to (3.4) in the non-modular case was found by Threlfall and Seifert, Topologische Untersuchung der Diskontinuitdtsbereiche endlichen Bewegungsgruppen des dreidimensionalen sphdrischen Raumes, Math. Annalen, vol. 104 (1931), pp. 1-70, for the special case where n = 3 and G is a finite group of rotations.

54 MOSES RICHARDSON

(3.6) Let R,(K, «) = 0 for all primes 7. Let N be a prime such that the number ¢,,(N) of m-dimensional coefficients of torsion of k divisible by N is not zero. By the well known relation

R,.(k, N) = Rak) + tn(N) + tra),

we have R,,(k, N) # 0. By (3.3), this implies p = 0 mod N. This proves that if R,.(K, ) = 0 for all primes x, then every m-dimensional torsion coefficient of k is of the form r{'r3? --- wi%, where m, m2, --- , t_ are the prime factors of p. In particular this conclusion is valid when K is an n-sphere for

m=1,2,---,n—1/

(3.7) Let K, be an n-circuit. If G contains a transformation which reverses orientation, then the totality of orientation-preserving transformations of @ forms an invariant subgroup of index 2. Thus half of the transformations of G reverse orientation. Now if R,(K,) = 1, then R,(k,) = 0, for (v7;) = (z7,) = (1—1+1—1.---) = (0). If, however, G has no orientation-reversing trans- formation, and if R,(K,) = 1, then R,(k,) = 1. If pis odd, then k is a cireuit for G has no orientation-reversing transformation and R,(k,, 2) = 1. If p is odd and K,, is a non-orientable circuit, then so is k,.

Part II

In this part we shall apply the results of Part I to find certain Betti numbers of symmetric products.

4. Direct and symmetric products. If A is a class of abstract elements, the direct q-fold product A X A X --- X A (q factors) is the set of ordered q-tuples (x X y X --- X z) where z, y, --- ,z are elements of A. The q-fold symmetric product of A is the set of non-ordered q-tuples (2, y, --- , 2) where x,y, -++ ,2 are elements of A. The symmetric q-fold product of A can evi- dently be obtained from the direct q-fold product by merely identifying the element (tx X y X --- X z)of AXA X --- X A with all its images under the group of permutations of the z, y, --- , z.

5. Subdivision of direct g-fold product complexes. Let K,, be a complex of simplexes in a cartesian space S,. The direct q-fold product K, X Ky, X «++ K Kn = Kan can be considered as being immersed in a cartesian space SxS xX--- X 8, = Sw,

and will evidently be composed of flat convex cells. Let any point X of K, have the codrdinates x, x2, --- , 2, in S,. The codrdinates in S,, of the point

® See note 8.

HOMOLOGY CHARACTERS OF SYMMETRIC PRODUCTS v0

(5.1) XXYX---XZ

of Kon are (x1, Xo, +++ 5 Loy Yty Yry *** 5 Yoy «+ * 5 Z1y 24 *** 4 2), OF in Condensed notation

(5.2) (z, J, °° , 2).

Now we take K,, as our basic complex, and a group isomorphic to the sym- metric group on gq letters as our group G: namely, the group which takes the point (5.1) with codrdinates (5.2) into all points obtainable by permuting the letters in these symbols. All the points of K,,, of the form

(5.3) XXXX+++XKX

are invariant under the gq! transformations of G, and the totality of them con- stitutes a point set homeomorphic to K,. Likewise the set J of points (5.3) contained in a cell of K,, of the type

(5.4) EX X Ei X +--+ XE}

is homeomorphic to FE; and only cells of type (5.4) contain points of the type (5.3). Thus J is a u-cell which we shall call E%', and the totality of these cells constitutes a complex homeomorphic to K, which we shall call K°.

The cells of K,, are merely permuted by the transformations T, of G. We shall now obtain a simplicial subdivision K‘}) of K,, which will satisfy the re- quirements (a) and (b) of §1 with respect to the T,’s. The process of sub- division begins with the 1-cells and proceeds to the m-cells only after all the (m — 1)-cells have been subdivided. The procedure for m-cells is as follows.

Let | o' |, | o? |, --- , be the (m — 1)-simplexes on the boundary of the m-cell E of K,n. Now: (1) if E is of type (5.4) we introduce an arbitrary point P, situated on the u-cell of K° which is imbedded in E, as a new vertex; then we replace E by the set of all m-simplexes | Po' |, | Po? |, --- , together with all the faces of lower dimension of these new m-simplexes except those on F(E); (2) if E is not of type (5.4), we introduce an arbitrary point P in E as a new vertex, and proceed as in (1); simultaneously we introduce the points

T)P (A = 0,1, --- ,p — 1)

in T\E as new vertices, and we subdivide the cells T,E£ similarly. The sub- division thus obtained is called K‘').

K“? satisfies requirements (a) and (b) of §1 with respect to the T’s.

Proof. (a) Any simplex | Z,,| = | VoVi --- Vm | is flat, and the T’s are evidently linear in the coérdinates of S,,. Thus T£,, is also flat and is there- fore completely determined by its vertices T,Vo, T,Vi, --- , T»V m.

(b) The vertices V and T,V of K‘') cannot be adjacent. (1) Suppose V and T,V are vertices of K,,. If they had been joined by a 1-cell, they were separated by the insertion of a new vertex on that 1-cell; and if there had been no such 1-cell, they are certainly not adjacent. (2) If V is not a vertex of K,,, then

56 MOSES RICHARDSON

neither is T,V. Now, when V and 7\V were introduced within the convex cells of K,, containing them, they were joined by new 1-cells only to the vertices of these containing cells; thus there can be no 1-cell of KY joining V and T\V. Now we have a simplicial subdivision xe of K,, which satisfies all the requirements of Part I. Each convex cell of K,, has been replaced by a set of simplexes which we shall orient concordantly with the original convex cell. In an m-chain, we shall have each m-simplex of } ein affected by the same co- efficient as affected the original convex cell of K,, previous to the subdivision. Evidently, AK®? = k,,, is the q-fold symmetric product of Ky. (5.5) The closure of each cell of K‘') is a subcomplex of K‘'). Let some of these subcomplexes be subdivided by section” into convex complexes. It is clear that we can further subdivide Fay into a complex of simplexes, say K‘’?,

which will again satisfy requirements (a) and (b) of §1.

6. Two-fold symmetric products. Let K, be a simplicial complex, Ky, = K, X K,, the direct product complex, KS!) the subdivision of K2,, and S, the space in which K,, is immersed ($5). The group G has the elements J, T where T is the transformation which interchanges the points X K Y and Y X X of Kan. To obtain the m™ Betti number R,,(k2,) of the 2-fold symmetric product com- plex ka, = AK$'), we have only to write down explicitly the matrix (x? ) = ("a:;) + (6;;) mentioned in (3.5) and to compute its rank.

Let E, = AoA; --- A, and E, = BB, --- B, be simplexes of K,. Let A; have the codrdinates (xj, 73, --- , x}), denoted briefly by 2‘, in the cartesian subspace S, of S, determined by Ao, Ai, --- , A». Likewise, let B; have the codrdinates (yj, y}, --- , yi), denoted by y’, in the subspace S, of S, deter- mined by the vertices of E,.

Lemma. T(E, X E,) = (— 1)" E, X E,.

Proof. The orientation of the cell Z, X E, is uniquely determined by the sign of the determinant of the homogeneous coérdinates (x9 = 1) of the vertices of the simplex"

(6.1) Ay X By Ai X Bo --+ Ap KX Bo Ao XK By --+ Ao X By. The orientation of E, X E, is likewise determined by the simplex (6.2) Bo X Ao Bi X Ao --- By X Ao Bo X Ai --+ Bo X Ap.

Now the transform T(E, X E,) of E, X E, has its orientation determined by the transform of the simplex (6.1), namely

(6.3) Bo X Ao Bo K Ai --+ Bo X Ap Bi X Ao---+ By X Ao.

The determinants of the codrdinates of (6.2) and (6.3) in S, & S, are, in con- densed notation,

10 Lefschetz, loc. cit., p. 67. 1! Lefschetz, loc. cit., p. 224.

an

Vv

HOMOLOGY CHARACTERS OF SYMMETRIC PRODUCTS 57

tT Ff Ts iy tis # 1 y? 2x and i; # is # | iy iv # ity. 2

We can evidently transform one of these determinants into the other by pg inter- changes of rows, which proves the lemma. An immediate corollary is the cor- responding relation for chains, i.e., T(C, X C,) = (— 1)"*C, X C,. By §5, this formula holds for K$).

Consider the set of cycles a} (i = 1, 2, --- , R,(K,)) constituting a minimal base for weak homology of s-cycles of K,. The set of cycles ai X a}_, form a minimal base” for weak homology of m-cycles on Ke,. If m = 2s, there are some m-cycles of the form ai X ai = A}, which are transformed by T into themselves, except perhaps for orientation, since T(a' K a‘) = (— 1)° a‘ X a’. These are evidently R, (K,) in number. All other cycles of the base are trans- formed by T into different cycles of the base. We can therefore rename the cycles of the base as follows:

™, re, — , I’, r, r, eal , Tv, At, A’, wat Si , A* (kK = R, (K,);h + 2g = Ra(Kon)), where fi = Tri, TA‘ = A‘ = (— 1)* A‘, and h = 0 unless m = 2s. We are

now able to write down the matrix of the weak homologies TT‘, = ="a,; C4, where we shall arrange the cycles in our matrix table in the order (6.4) horizontally and vertically beginning with the upper left hand corner. If m # 2s, we have

0] ("a;;) = ( ), I 0

(6.4)

and if m = 2s,

0 J 0 ("a;) =|I 0 0 , 0 0 (-—1)'I’

where J and J’ represent identity matrices. Bearing in mind the orders of these submatrices, we compute easily the rank of the matrix (27;) = ("ai;) + (4), which, by (3.5) is equal to R,, (ken). Thus,

® Lefschetz, loc. cit., p. 228.

58 MOSES RICHARDSON

THEOREM 2. The non-modular m-dimensional Betti number of the 2-fold sym- metric product ke, of K,, is given by

(4 Ry, (Ken) (m # 2s) (6.5) Rn (ken) = ‘4 [Rm (Ke,) — R, (K,)] (m = 2s, s odd) (3 [Rn (Ke) — R,(K,)] + R,(K,) (m = 2s, 8 even)

where Ko, = Ky, X Ky.

Since p = 2, it is easily seen that formulae (6.5) give the Betti numbers mod nm” provided x is an odd prime, where all Betti numbers in the formulae are understood to be mod x“.

Examples. (1) Let K, be a 1l-sphere H;. For H, we have Ry = R, = 1. For the torus H; X H, we have Ry = R. = 1, Ri; = 2. For the Mébius strip A(H, X H,) we have Ry = 3 [1 — 1] +1 =1,R, = 3 [2] =1,R,=3[1 —1)=0. (2) Let K, be a 2-sphere H,. For H. we have Ryo = R. = 1, Ri = 0. For He. X Hzwe have Ry = Ry = 1, Ri = R; = 0, R, = 2. For the complex projec- tive plane A(H, X H.) we have Ry = R, = R, = 1, Ri = R; = 0.

7. Two-fold symmetric product of a circuit. First, note that after the subdivision of §5, we can name the m-simplexes of the complex K$') as follows: D}, i i Ni » Di; D}., Dy.., aby ia , Dk, . i's ee Ss where D;. = TD,, and D°* denotes a simplex of the subecomplex K°. The bounding relations of these simplexes can be written as follows:

Dj, > 2 i; Diy + 2 nj, D3 + B55 DAL, (7.1) Dy, > 2 nj Day + Deis Dl + DEG Dr, Di‘ — = 65; Dri, where the coefficients can take only the values + 1, — 1, or 0 and where not both ¢;; and n;; are + 0. On k2, we have ‘the bounding relations dj, > = (ej + ni) hy + Doan , dm’ > 2 Oidmis where di = A Di and d®% = A D", the d being cells of k& = A K°. Now let K, be an absolute n-circuit. The direct 2-fold product Ke, is an absolute 2n-circuit which is orientable if and only if K, is orientable.” TueoreM 3. If K,, is an absolute circuit, then, for n = 1, kon is a relative 2n-

circuit modulo its subcomplex k°, and, for n > 1, ken is an absolute 2n-circuit. Proof. On the subdivided complex KS'), we have

> F(D;,) + X F(D;,) = 0 mod 2.

(7.2)

13 Lefschetz, loc. cit., p. 231.

HOMOLOGY CHARACTERS OF SYMMETRIC PRODUCTS 59

By (7.1) this becomes a (Ce; + nis) D3. + (ej + nu) Di, + 26; D2J_,] = 0 mod 2. t]

Hence, Dy Fai.) = Do (les + m4) dda + Si d2h-al = DL gi d2i_, mod 2, . 1) td”

where evidently the cells d}/_, exist only when 2n — 1 S n;i.e., when n = 1." Thus k>, is an absolute 2n-cycle mod 2 for n > 1, and a relative 2n-cycle mod 2 modulo k° for n = 1.

We have yet to show that no subset of the 2n-cells of k., forms a 2n-cycle mod 2. Suppose there were a subset d},, ---,d3, (¢ < mu + v) such that

> F(di,) = 0 mod 2, ie.,

i=1 Pe [Ces + ni) dh. + Si d24_,] = 0 mod 2, t)

then it would follow that

Do F(A’ d;,) = F(D}, + --- + Di, + Di, +--+ + Dj,) = 0 mod 2,

i=1 which would contradict the hypothesis that Ke, is a circuit. This proves the theorem.

TueoreM 4. If the absolute circuit K, is non-orientable, so is kon; if Ky is orientable, then kz, is orientable or not according as n is even or odd.

Proof. If K, is non-orientable, R,(K,) = 0, so that, by (6.5), Ren»(ke,.) = 0. If K,, is orientable, R,(K,) = 1, so that, by (6.5), Ro.(ke.) = 1 or 0 according as n is even or odd.&

Example. The Moébius strip is an example of Theorems 3 and 4 for n = 1. The “edge” of the strip is k}. The strip is a relative 2-circuit modulo its edge, and is non-orientable.

8. Three-fold symmetric products. Let K, be a simplicial n-complex, K;, = K, X K, X K,, the direct 3-fold product, K') the subdivision of Ks, which must be made in order to satisfy the requirements of Part I, and ks, = AK) the 3-fold symmetrie product. Here G is of course isomorphic with the symmetric group on 3 letters. We have

ToE,XE,XE, = E,XE,XE,, T:E,XE,XE, = (— 1)" E,XE,XE,, (8.1) T.E,XE,XE, = (— 1)"**" E,XE,XE,, T:E,XE,XE, = (— 1)" E,XE,XE,, T.E,XE,XE, = (— 1)” E,XE,XE,, TsE,XE,XE, = (— 1)?*+er+ E,XE,XE,.

4 The case n = 0 is obviously devoid of interest.

1% Cf. (3.7).

60 MOSES RICHARDSON

The exponents of —1, which we shall call the orientation exponents of the transformations (8.1), are obtained by methods analogous to those of the Lemma of §6.

For example, let us obtain the orientation exponent of 7), say. Let E, = AvA, --- Ap, E, = BoB, --- By, E, = CoC; --- C,, and let the codrdinates of A;, B;, C. be, in condensed notation, zx‘, y’, 2*.

The simplex determining the orientation of E, X E, is

Bo X CoB: X Co --- By X CoBo X Ci --+ Bo X C;; thus the simplex determining the orientation of E, X E, X E, is (8.2) Bo X Co X AoBi X Co KX Ao -++ By K Co X AoBo X Ci XK Ao: Bo X C, X AoBo X Co X Ai +++ Bo KX Co X Ap.

The simplex determining the orientation of T;E, X E, X E, is the T\-transform of the simplex which determines the orientation of E, X E, X E,. Writing out the latter simplex, and applying 7) to it, we find the desired simplex to be

(8.3) Bo X Co X AoBo X Co XK Ai «++ Bo KX Co X ApBi XK Co XK Aoss B, X Co X AoBo X Ci X Ao +++ Bo KX Cy XK Ao.

The determinants of the codrdinates of (8.2) and (8.3) are

ly & # Lif #& # ist # if #f # tis & £ ir *& iF #t # it e#

: and . iy vr # is # # iy ft # Lr et # iyrtrt # irr

Since these determinants can be transformed into each other by pq + pr inter- changes of rows, we have the desired orientation exponent.

The relations (8.1) hold also for chains of KS'). Now we consider the set of m-cycles a‘ x ai X a‘(p + q + r = m) which constitutes a minimal base for weak homology of m-cycles on K%'). Since the formulae (8.1) tell us the precise effect of any 7, on any cycle of this base, we are able to write out the matrix (x7;), and to compute its rank, as we did for the 2-fold symmetric product.

eS ff «ea an = a 6a at Oe ok fe ae Oo

Pm)

wre FR

HOMOLOGY CHARACTERS OF SYMMETRIC PRODUCTS 61

We omit the details of this process. The result is as follows: THEOREM 5. The m-dimensional non-modular Betti number of the 3-fold sym- metric product ks, of a simplicial complex K,, is given by

4Rn(Ksn) — Ri(K,) — Rs(K,) — +++ — Ri(Ka) (m ¥ 3), Rn(ksn) = 3[Rn( Kon) — R.(K,)] + RA(K,) = Ri(K,) = R(K,,) - -) > 2 RAK,) (m = 3s),

where l is the highest odd integer = m/2.

Since p = 6, the same formulae yield the Betti numbers mod x“ provided x is a prime > 3, where now all Betti numbers in these formulae are understood to be Betti numbers mod rx“.

Example. Let K, be a l-sphere H;. On K;, = Hi X H, X H, we have

Ro = Rs = 1, Ri: = R2 = 3. Onks, = ACA: X Hi X H,), we have Ro = R, = 1, R. = R; = 0. 9. q-fold symmetric products. Consider a cell of the form (9.1) E.XE.X-::-XE, (q factors)

Let us call any transformation which interchanges two adjacent cell-symbols in (9.1) an “inversion.’”’ For example, the transformation that takes (9.1) into E, X E, X --- X E,isaninversion. From a consideration of the structure of the determinants involved in the computation of the orientation exponents (§8), it is easily seen that any inversion applied to (9.1) will introduce a factor of — 1 with an exponent equal to the product of the dimensions of the cell-symbols interchanged. Since any transformation of the symmetric group on q letters can be produced by successive inversions, its orientation exponent can be deter- mined. It seems to be impossible to write a general expression for these ex- ponents in terms of q; the difficulty of generalization seems to be at least partly group-theoretic. However, for a given integer g > 1, we can write out the transformations of the symmetric group on q letters, and compute the orienta- tion exponents of each. These exponents give us exact knowledge of the effect of each T, upon any cycle. Now, given a definite complex K,, we can write

down the cycles of a base for = of m-cycles on K,,. Then we can write out the -1 q! matrices ("a?;) and compute the rank of (z7;) = } ® ("a*;). Thus, we can X=0 find the Betti numbers of the g-fold symmetric product k,, for a given K, and a given q. Example. Let K, be a 1-sphere and gq = 4. Then on Ky, we have

RaRelLhoh=¢4h oF

62 MOSES RICHARDSON

It is easy to pick out a basal set of cycles for each dimension. Computing the orientation exponents of the 24 transformations of G, writing out the 24 matrices ("a*;) for each dimension, and so on, we find that on ks, we have

Ry = Ry = 1, Ry = Rs = R, = 0. Part III

We shall now determine the mod 2" Betti numbers R,,(k2,, 2") of the 2-fold symmetric product ke, foru > 1. All homologies and equations in the following sections will be understood to be homologies and congruences mod 2", u > 1. Unless otherwise indicated, all cells, chains, etc., will be of dimension m.

10. Let W', W?, --- , W" be the cycles of a base for weak homology of m- cycles and Z', Z?, --- , Z*, the cycles of a base for m-dimensional zero-divisors of a given complex K.

If C', C?, --- , D', D®, --- , ete., are chains of the same dimension, the symbol [C, D, ---] shall denote a linear combination of these chains with integer co- efficients not all zero. If the coefficients of such a linear combination are either 0 or 1, not all zero, we write {C, D, ---}. If the coefficients of a linear com- bination are allowed to be all zero, we write [C, D, ---]’ or {C, D, ---}’. If the same symbol appears more than once in an argument, it does not necessarily represent the same linear combination at each occurrence.

(10.1) There can exist no relation of the form 2X ~ {W} + [Z]’.

Proof. Certainly X ~ =A,W‘ + [Z]’. Let t be the l.e.m. of the orders of the Z’s. Since our modulus 2" is a power of a prime, it is easily shown that ¢ is not congruent to zero. Now

(10.2) 2X ~ 2t(Ai\W'! + A.W? + --- + A,W’).

Suppose there were a relation of the form 2X ~ = a,;W‘ + [Z]’ where the a’s are 0 or 1, not all zero. Then

(10.3) 2X ~ t(a,W' + @W? + --- +a4,W’). Hence 2¢ ~ 0. From (10.2) and (10.3) we obtain 0 ~ (2A, — a)W' 4+ --- + (2A, — a,)W’.

Therefore ((2A; — a;) = 0 for all values of 7 But some a; = 1. Therefore some 2A; — a; is odd, and t must be congruent to zero. This is a contradiction. If there are no zero-divisors on the given complex, the proof is essentially the same.

11. From this point on all chains, ete., shall be of Ke, or the subdivision Ky, (Part II) unless otherwise stated.

If a chain has no cells of K°, we shall attach an asterisk to its symbol, as X*. If a chain has only cells of K°, we attach a zero to its symbol as X°.

16 A chain is said to “‘have’’ a cell if the cell occurs in the symbol for the chain with a non-zero coefficient.

HOMOLOGY CHARACTERS OF SYMMETRIC PRODUCTS 63

If TX = X = eX, |€| = 1, the chain X will be called invariant. Sometimes an invariant chain will be called positively or negatively invariant according as e=-+lor-—1. If X = X, then X is of the form X* + X* + X°; andif X¥ = —X, then X is of the form X* — X*. If H —C, where H and C are both posi- tively or both negatively invariant, we write C ~ 0. These special homologies obey the same rules as ordinary homologies.

(11.1) There can exist no relation of the form L+L+ {W} +[Z ~0,

where the cycles W%, Z% form bases, as in §10, for K°. Proof. Suppose there were a relation

(11.2) H+H+H->L4+L 4 {W% 4+ [Z) ~0. In any case we can write L + L = L* + L* + 21°, so that (11.2) becomes (11.3) H+H+ H°>L* + L* 4+ 21° 4+ {Ww} + [Z.

Let H + H — L* + L* + X°, say. If a cell of K° occurs in F(H), it occurs in F(H) with the same coefficient, so that X° = 2Y°, say. Therefore (11.4) H+H—L* + L* + 2Y°. Let L®° — Y®° = U®. Then from (11.3) and (11.4) we have H® + 2U® + {W®} + [Z°)’ ~ 0,

contradicting (10.1). (11.5) IfH+H—-C + €C,|\e€| = 1, then F(C) ~0.

Proof. Let L = C — F(H). ThenL—F(C). Now

F(C) + €F(C) = F(F(H + eff)) = 0. Thus F(C) is invariant. Also L+h=C+ceC — (F(H) + €F(A)) = 0.

Hence L is invariant. Therefore F(C) ~ 0. (11.6) IfC + C ~0O, then the cycle AC either is ~ on kz, to a zero-divisor of k°, or is ~ 0 on kon.

Proof. By hypothesis, H* + H* + H®+~C + C. Let H* —-C — Dand let H#® > J* Then H* + A*4+ H°5C4+C-(D4+D)D)4%=C4+ 6. Hence J° = D + Dand 2AD = AJ®. But AH® > AJ® ~ 0 on k°; therefore

2AD ~ 0 onk°. Thus AD either is a zero-divisor of k° or is ~ 0 on k°. But AH* — AC — AD. Hence AC ~ AD on ke, which completes the proof.

12. Let ai = DiiE* (i = 1, 2, --- , R,(K,)) be the cycles of a base for weak homology of s-cycles of K,. Then

43,7 @; Xai = Du ES xX Ef = Do this Es X Ef + D> (ti)*Es Xx ES. a,p ax a

64 MOSES RICHARDSON

(12.1) Associated with each A}, (i = 1, 2, --- , R.(K,)) there exists on KS}, or a suitable subdivision of K}') a sequence (‘A}!, ‘A3!_,, --- , ‘A2*) of cycles and a sequence (‘X},, ‘X3,-;, «++ , *X*,,) of chains such that

‘Aj’ = X5 + EX}, |e] = 1; FOX}) = Aj, (fj = 2s, 2s — 1,---, 8 + 1)

and ‘A?* = Wi + Wi + W2', where the cycles W°* form a base for weak ho- mology of s-cycles of K°.

Proof. Let us drop the superscript 7 temporarily and consider a definite As,. Let E* X E% be an invariant cell of Ke, occurring in the symbol for A:, with a non-zero coefficient. Let S}, be the flat subspace of S., (§6) which contains E* X E% and let (xf, xo, --- , 2%, yf, yz, --- , yZ) be a cartesian codrdinate system in S$, so chosen that T interchanges the points (z, y) and (y, x) of Sf,. Let | yf, | be the set of non-oriented 2s-simplexes by which the subdivision K‘') replaces the non-oriented convex cell | EY X E% |. Thespace Sf,_,, defined by the equation xf = y{, contains all the invariant points in the closure Cl | ¥f, | of |¥e, |. It is clear that S},_, subdivides | ¥, | by section into two sets of non- oriented convex cells, | ¢f, | and | $f, |, which are interchanged by T and are such that the closure of each set is the closure of a 2s-cell. Do this for every a. By (5.5) we can further subdivide K%') into a complex KG) of simplexes such that conditions (a) and (b) of §1 are satisfied with respect to T. Let yf, and ¢$, be the chains obtained from | yf, | and | ¢f, | by taking the chain sum of their simplexes oriented concordantly with Ef X E%. For the sake of definite- ness, let s be odd. Then yf, is negatively invariant,” and ¥{, = $3, — $%,. Now let

Xt. = > tote Bt x 2+ 6 os. . a<s Then Ax, = X}, — XJ, = A3’,say. Since Cl | $7, | is the closure of a 2s-cell, F(¢,) is a simple circuit. Thus, in particular, the 2s-simplexes of F(@¢=,) in Sf,—, oecur with the coefficient one; hence, they occur in the symbol for F(X},) with the coefficient #2. Let ¥f,_, = F(@f,) and let | ¥f,_, | be the set of the non-oriented simplexes of ¥f,_,. The space S{,_., defined by the equations z{ = y{, #] = yf, contains all the invariant points in Cl | ¥%, | and subdivides | ¥f,_, | by section into two sets | @f,_, | and | @$,—, | of non-oriented convex cells, which are interchanged by 7, and are such that Cl | ¢{,_, | is the closure of a (2s — 1)-cell. Introduce this subdivision for every a. By (5.5), we can again subdivide KS) into a com- plex K*) of simplexes so that conditions (a) and (b) of §1 are satisfied with respect to T. Now, let ¥f,-, and ¢$,-, be the chains obtained from |¥F,—, | and | $f,_, | by taking the chain sum of their simplexes properly oriented. Since F(X3,) — F(X},) = F(A?!) = 0, F(X.) is positively invariant. Let F(X%,) = A2%_,. We have F(X3,) = r(d ta ts EZ X E*) 422... a<B

17 If s is even, ¥{, is positively invariant and yf, = #5, + ¢],- The proof proceeds with no essential change.

o~ Fe & 8 . 3 BRA be

HOMOLOGY CHARACTERS OF SYMMETRIC PRODUCTS 65

Since yf ,_, is invariant and A}{_, is positively invariant, ¥{,_, must be posi- tively invariant. Thus we can write

Aj i-s = ‘Se~t + se + ze ($3 5-1 + $3 »—1) °

ket Xt. @ Ve. + BE ..., oo et Ot. @ Fe. + Zee. Se Cl! ¢f,_, | is the closure of a (2s — 1)-cell, F(¢f,_,) isa simple circuit. There- fore all the (2s — 2)-simplexes of F(¢?,_,) contained in S{,_, occur in the symbol for F(¢f,_,) with the coefficient one. Hence they occur in the sym- bol for F(X} ,_,) with the coefficient t2. We continue this process until we have

2s 7* v* Av+i = Xi+1 = Xo+1 .

The space S*, defined by the equations rf = yf (q = 1, 2, --- , 8), contains all the invariant points in Cl | yf, | and subdivides | ¥%,, | by section into two sets | ¢7,, | and | ¢%,, | of non-oriented convex cells which are interchanged by T and are such that Cl | ¢%,, | is the closure of an (s + 1)-cell. We introduce this subdivision for every a. By (5.5), we can further subdivide our complex into a complex K$') of simplexes which satisfies (a) and (b) of §1 with respect to T. Now, F(¢%,,) is a simple circuit. Thus, in particular, all the cells E°* (or the smaller simplexes by which they have been replaced during the successive sub- divisions) of K° which are on F(¢%,.,) occur in the symbol for F(¢%,.,) with the coefficient one. Hence they occur in the symbol for F(X*,,) with the co- efficient ¢2._ Thus we can write :

F(X041) = X5 + 304 202 E* = 43°, say.

But > #2E°* = > (2 — t.)E°* + 2t,.E°*. Since t2 — t, is even, we can write 22 EF°* =U°+0° +25. E°*. LetW, = X* + U® and W? = 51, E**. Then A?* = W, + W, + W°. The symbol for W° has the same coefficients as that for a,. Since K, and K° are identical in structure, the cycles W°* form a base for weak homology of s-cycles of K°.

(12.2) The cycles Aj* (s = 0,1,---,n;h = 8,8+1,---,28 — 1) of (12.1) satisfy the relation 247° ~ 0.

For, X}., — A?° by definition, and 43° = ¢ A?*,|«| = 1. Therefore,

Xie + €Xh41 — 2A; ~0.

Consider now a base for zero-divisors of dimension m of Ko,, like that of Lefschetz (loc. cit., p. 229) remembering that here Ke, = K, X Ky. Recalling the properties of our transformation 7’, it is easily seen that the cycles of this base can be renamed 2‘, 2! = TQ‘, and 6‘, where TO‘ = @',|«| = 1. The O’s are the cycles of the form bi x bi (m = 2s), where ci,, — ¢' b! on Ky.

(12.3) Associated with each cycle 0}, of Kon, there exists on Kn, or a suitable subdivision of Kon, a sequence of bounding cycles (‘03', ‘O3%_,, --- , 02") and a sequence of chains (‘3 ,, ‘£2,-,, --- , €*4,) such that, for each value of i, we have

‘97° = ¢} + € €F,|¢| = 1; FC) = ‘97! (j = 2s, 2s —1,---,8+1)

66 MOSES RICHARDSON

and ‘92° ~ ¢' Z°* on K®° where the cycles Z°' constitute a base for zero-divisors of dimension s of K°.

Proof. We drop the superscript 7 temporarily and consider a definite 02, = b, X b, where c,41 — £ 6,. Consider the bounding relation

Cot X Co > F b, XK Ceti + (—1)** € Cs41 X a”

For the sake of definiteness let s be odd. Then ¢,41 X ¢s4: is positively in-

variant and we can write ¢,41 X ¢s41 = Ne42 + fee+a- We can choose 73 ,.. * fon

so that 3.42 — £0. X Cour — Yoe41- Since

F(n3 +2) + F (RE, +2) = F(Cs41 x Co41) ’ we have -* * i Neste 7 § Coun X OD. + View :

Thus Y},,, is negatively invariant and can be written as 93,4; — 2.11. Now F(¥3.41) = F(gb. X ess) = &b, X b, = OF%, say. Evidently 03: is nega- tively invariant, and can be written as £, — §&},. Choose 3,,, so that then > 8, — YEs- Since Forts.) — Flv) = 8, — Gey we have gst at . - Y3,. Hence Y3 »_ is positively invariant and can be written as "2s + %2.- Now F(Y3,) = P(e.) = 03°_,, say. Since 05 °_, is positively invariant, we write 02$_,= &,-, + £3,_-,. Choose 73, so that

nos — $3 4-1 = Weoue: Since F(n2.) + F(z.) = &4-1 + &,-1, we have ins S.- » + ¥5.-+:

Thus Y},_, is negatively invariant. We proceed in this way until we have Yes = ne+2 — fe+2 — F(E,) = 9 ~ = 0741, say,

where 7*,, is chosen so that n*,. — &*., — Y,4,. Now at4. 8%, — Yyay. Thus Y,,, is positively invariant. Finally we write Y,,, — F(é*,,) = 02°, say. Let us trace the effect of this process upon a single convex cell E%,, K E%., occurring with a non-zero coefficient in ¢,4; X Csi, where ¢,4; = StaHS,,. The totality of invariant points in Cl| Z%,, xX E%,, | constitute the closure of an (s+ 1)-cell. Now, the separation of Y;, into n, + e;,|€| = 1, can be accom- plished by subdivision by section by means of flat (kh — 1) spaces S¥_,, as in (12.1). Now Cl| Y,_1 | is seen to be composed of points common to Cl | F(n;) | and Cl | F(#;,) |. Hence Cl | Y ott | consists entirely of invariant points; there- fore Y,4; can be denoted by Y°,,. Now, Vows = Dt2E°%,, by the same argu- ment as in (12.1). Then Y°,, = V°,, + V°4, + Dt.H°%,, asin (12.1). We have therefore F(Y°,,) = 02° = 2F(V°.,) + F(2t,.E°%,). But the chain symbol for F(2t.£°%,) has the same coefficients as that for ¢b,, and can be de-

18 Lefschetz, loc. cit., p. 227. 19 Tf s is even, the proof is essentially the same.

HOMOLOGY CHARACTERS OF SYMMETRIC PRODUCTS 67

noted by ¢Z°. Since K, and K°® are identical in structure, the cycles Z°* form a base for zero-divisors of dimension s of K°. Evidently, each ‘62° ~ ¢'Z°'.

The cycles ‘Aj* arising (12.1) from the cycles A},(s = 2n, 2n — 1,---; i = 1,2, --- , R.(K,)) will be denoted generically by A,. The cycles ‘03° aris- ing (12.3) from the cycles 0; , (s = 2n, 2n — 1, ---;7 = 1,2, ---) will be denoted generically by 6, except for h = s.

13. We shall need the following lemmas. (13.1) If D + «D, || = 1, isa cycle, thn D+ D~ [Pr + Ff, A,2 4+ &, 0)’. Proof. Let « = 1. We have (13.2) D+Dw~s(aTi + af) + [A]’ + 202‘ + 6,85 + [oy’. Applying T and subtracting the resulting homology from (13.2) we obtain 0 ~ S(a; — a,)(T* — TP) + [A]’ + 2(b; — 6,)(@' — &) + [oy

Hence a; = 4;. Also either b; = 6; or b; — 6; is a multiple of the order of 2°. In the latter case, we have (b; — 6,)Q‘ ~ 0, or 6,8‘ ~ b,Q‘. Thus in either case we can rewrite (13.2) as

D + D~ a(t + TM) + [Ay’ + =b(Q* + 2) + [ey’. This completes the proof. If « = —1, the proof is essentially the same. (13.3) If Dn + eDm, | €| = 1, is a cycle, then Dm + Dm > [Pm + Fm, Amy Qn + my Onl’ + [An}’ + [On]? + 21Z 0)’,

where no A,, occurring with a non-zero coefficient is of the form Wn + Wm + WS.

Proof. By induction.

(A). By (13.1), the theorem is true for m = 2n, since for this highest dimension cycles of type A, 6, and Z° do not exist, and ~ means only =.

(B). Assume the theorem for the dimension m + 1 S 2n. Case I: let e = 1, m = 2s,s odd. By (13.1), there exists a chain H,,,, such that

(13 4) Has —— D,, aa Dz, a is ab m Beas Qy + Dn, = a Since s is odd, A,, and 0,, are negatively invariant. Ifence Ans — Bass =“ ZAn, Onl’.

But 2{A,,, 0,,]’ must be zero for otherwise we would have 2[A,,, 0,,.]’ ~ 0 which is impossible since the A’s and 0’s are elements of a base for homology. There- fore Hy4.1 — Hy +: is a eyele, and by the induction hypothesis

|; ewe —_ Anas =~ [T'42 aa | Qmi1 — mst) + {Amsi}’ + [Omsi)’ + 2Z 241)’

since there are no cycles As,.; Or O2,4;. None of the cycles A,,.; or 0,41 0c-

(13 .5)

68 MOSES RICHARDSON

curring with non-zero coefficients in (13.5) can be of the form X + X, and 2[Z>, +1] = 0, since the terms of (13.5) must be negatively invariant. Therefore | = Aart - ; aa and Omit = Emit = E41 Now let

Cir = [Pmsty Qmail’ + {Xmai}’ + [Email],

where the coefficients of the terms on the right are the same as in (13.5). Now (13.5) becomes Hingr — Aina ~ Coir — Emr, OF

(H mt — Cm) — (Amit — Coit) ~ 0. By (11.5), F(H naa — Cm) ~» 0, or F(H mys) ~ F(Cms)- By §12, F(Xms) = A, and F(Enss) = 0, or ¢Z°. Thus, (13.6) Dm + Dm + [Pm + Bry Amy Qm + Gy Onl’ [Anl’ + [Om]’ + [Znl’.

By (12.2), [A,,]’ can be replaced by {A,,}’. Therefore, no A,, occurring in (13.6) with a non-zero coefficient can be of the form W,, + W,, + W%, for otherwise, remembering that the cycles W2‘ form a base for weak homology on K® (12.1), we would have a contradiction of (11.1). There remains to be proved only that the expression [Z°]’ in (13.6) is of the form 2[Z°]’. We can write (13.6) in the form Cn + Cn + [Z°]’ ~0. That is,

Vers + Vans + Vor 7 Cn + Cn + (Z8)’ = CE + Ce + 2C8 + (ZS).

Now let Vo., — LE + LS. Thus Va, + Var, ~ L + Te + 20. Let V°,, ~J°. Now, 21° + Jo = 209 4+ [Z°)’. ButJ2 ~O0onK’. Thus 2L° — 20° ~ [Z°)’. Let L2 —C2 = U2. Now

2uU. _ [Z°]’ = AiZ°' 4- AZ)? + ee.

But the Z°’s form a base for zero-divisors on K°, by (12.3). Therefore the A’s must be all even, for U2 ~ =B,Z°' so that 2U°, ~ 22B,Z°' ; hence 2B; = Ai. Thus [Z°)’ is of the form 2[Z°]’. This completes the proof for Case I. The other cases (namely: « = —1, m = 2s, s odd; « = +1, m = 2s, s even; and ¢ = +1, m odd) present no new difficulties and are proved in essentially the same way.

14. Let v5 = ATi. If m = 2s, we have Ai, = Xi, + Xi, according as sis even or odd. Let 63, = AX}, if sis even, and let 63, = O if s is odd. (14.1) If sis even, 53, is a cycle and is not null.

For Xi, — ‘A25_, = ‘X},-, — ‘X},_, by (12.1). Therefore

AXi, — A‘A?t_, = 0.

Thus 53, isacycle. Furthermore AX}, is not null. For, suppose AX}, = 0. Then A’AX}, = A}, = 0, a contradiction.

HOMOLOGY CHARACTERS OF SYMMETRIC PRODUCTS 69

(14.2) The cycles y‘ and those cycles 5‘ which are not null constitute a minimal base for weak homology of m-cycles of kon.

Proof. The cycles y‘, 6‘ # 0 are linearly independent with respect to ho- mology. For, suppose there were a relation Zt;y‘ + si‘ ~ 0. Applying A’, we have Dt,(f‘ + I) + =s,A‘ ~ 0 which contradicts the hypothesis that the cycles I‘, I’, A‘ form a base on Kop.

There remains to be proved that every m-cycle d of ke, satisfies a relation of the form d = [y, 4]’. By (13.3),

Ad =D+D~([0 +T,A4,2 +S, 0)’ + {A}! + [6)’ + 212, where A, A, 0, 6 can be written as X + X,2+ %,¥Y + Y,y + 9 respectively. Let C= D+ (ll) + (X) + [oy + (YY) + tz)’ + yl’ + (ZY. Evidently C+ C ~ 0. By (11.6), (14.3) d ~ [y]’ + [6]’ + [AQ)’ + [AY + {Az}! + [Ay]! + [AZ*)’ + 2°,

where 2 is either null or a zero-divisor of k°. It is easily seen that all the terms on the right of (14.3) are = 0 except the y’s and the 4’s. Thus d = [y]’ + [6]’ which completes the proof.

Thus, R,.(k2n, 2"), u > 1, is the number of cycles y‘ plus the number of cycles 5‘ not null (14.1). Therefore,

THeorEeM 6. The numbers Ru»(ken, 2"), u > 1, are given by formulae (6.5) where all the R’s are now understood to be Betti numbers mod 2“.

BROOKLYN COLLEGE.

CONCERNING CERTAIN REDUCIBLE POLYNOMIALS By H. L. Dorwart

1. Introduction. In a joint paper' by Oystein Ore and the author, it has been shown that polynomials with rational integral coefficients which take the values + p (p arational prime) for m integral values of the argument must take the same value +p or —p for m > 5, and consequently have the form

(x — ay) (2 — ay) «++ (@ — Gn) h(x) + p. In the same paper it has also been shown that integral polynomials of the form (1) f(z) = a(@ — a) --- @—a,) +p

for n > 6 are irreducible in the rational domain if n is odd, and if n is even they may have only two factors of the degree n/2.2 A new proof of this result is contained in a recent paper by A. Brauer.* In this paper, Brauer also raises the question whether or not these reducible polynomials can exist for every even n. A numerical example of eighth degree for the prime 2879 has been given by Pélya,* and one of tenth degree for the prime 10079 is contained in Brauer’s paper. However, neither of these writers has obtained any general results on the subject.

We shall here find an expression for the necessary and sufficient conditions for the reducibility of these polynomials. From this result follows first the theorem of Brauer that the sum or difference of two such factors is a constant. Furthermore, it reduces the problem to well known problems in Diophantine equations for which partial solutions exist, and these solutions in turn give

Received by the Editors of the Annals of Mathematics February 27, 1934, accepted by them and later transferred to this journal. Presented to the American Mathematical Society, December 26, 1933.

1H. L. Dorwart and Oystein Ore, Criteria for the irreducibility of polynomials, Annals of Math., (2), vol. 34 (1933), pp. 81-94.

2 That this theorem does not hold for n = 6 is shown by the following class of polynomials

x(x + 1)(x — 1)(x — 2)(x — a)(x — B) — p = [x? — x — 1] [xt — 2x3 — (p+ 1)z2?+ (p+ 2) 2x4 p])

for primes of the form a? — (a + 1), where a is a positive integer > 2and8 = 1 — a.

3 A. Brauer, Bemerkungen zu einem Satze von Herrn G. Pélya, Jahresber. Deutschen Math. Ver., vol. 43 (1933), pp. 124-129.

‘Georg Pélya, Verschiedene Bemerkungen zur Zahlentheorie, Jahresber. Deutschen Math. Ver., vol. 28 (1919), p. 40.

70

CONCERNING CERTAIN REDUCIBLE POLYNOMIALS 71

various new decompositions for our polynomials. The new form of the problem also seems to indicate that decompositions exist for arbitrarily high degrees.

2. Necessary and sufficient conditions. Since we are considering only even degrees, let n = 2m. Next, let k(x) be a factor of f(x) which takes +1 for

M,-++,@m. Form > 5, it will have the form k(x) = b( — a) --- (x — ay) +1. But k(x) must take +p for a,4:, --- , @,, and the conditions to be satisfied are b(a; — a) --- (aj — an) = +p — 1 ({=m+1,---,n). The other factor, say I(x), must take +p for a, --- , dm, ie.,

U(x) = c(x — a) --- (© — ay) + p, and must take +1 for @n4:, --- , @n. Hence the conditions c(a; — a) --- (@i — an) = 41 Fp (i =m+1,---,n)

must also be satisfied. These conditions can exist simultaneously with those for k(x) only if b = ¢ = ~/a, and if the minus sign is chosen in front of the 1 and the signs in front of the p are reversed, i.e., k(x) and I(x) must take the values

| ay, ee a Am Am+ly = Ss a,

k(z) | +1 3 +p(—p) n U(x) | —p(+p) —1

and the reducibility conditions are

Va(ai — a) --- (a; — an) = +p — 1, (i =m-+1,---,n).

A similar situation results when it is assumed that k(x) takes —1 for ay, «++ , Gm ete. Hence we have the necessary and sufficient reducibility conditions for m > 5,i.e.,n > 10. It can easily be shown that these conditions also hold for n=10. For this degree, as stated in the introduction, decomposition is possible only in two factors of equal degree. Each factor must take +1 five times and +p five times. However, it has already been shown’ that it is not possible for such a factor to take five values +1 unless it takes only either +1 or —1 five times, in which case we have the situation discussed above. We can there- fore say:

THEOREM 1. The necessary and sufficient conditions for the reducibility of the polynomials (1) of even degree n = 2m forn > 8 are (2) Vala; — a) --- (a; — a) = |p +1] ((=m+1,---,n),

5 Dorwart and Ore, I.c., p. 85.

72 H. L. DORWART

and the decomposition can be made in any one of the four equivalent forms [Va(z— a) «++ (@ — an) + [Vale — a) --- (@ — an) Dp) [Va(z — a) «++ (@ — am) + Il Vale — anys) «+» (@ — as) F 1) [Va(x — amy) +++ (@ — ax) F piiva(e — a) --- (@ — an) + ppl [Va(x — amyi) +++ (@ — an) F pilVa(z — anys) +++ (@ — an) F 1),

with proper attention to signs.

3. Equivalent problems. From Theorem 1, evidently a must be a perfect square and +/a must be a divisor of |p +1]. Let|p+1|= Va-d. Then the conditions (2) reduce to

(3)

(4) (ai—a@)---(@i—an)=d (( = m+1,---,n). Let o1, o2, --- ,@m be the elementary symmetric functions of a, --- , dm, and pi, p2, +++ , Pm be the elementary symmetric functions of @n41, --- , Gn. Then conditions (4) become

(5) at — 0,0; ' +30; ° — --- + (—1)" 6, —d = 0

(i =m-+1, seam), which are easily seen to be equivalent to

(6) O11 = Py C2 = Py +++ 5 Oma = Pm-ty— | Om — Pm | = A.

For a set of integers satisfying the first m — 1 of these conditions, the last one determines d, which in turn for a given ~/a determines p. In fact, a well known theorem of Dirichlet® states that there will be an infinite number of primes p available.

A new wording of the first m — 1 conditions of (6) gives rise to

TueoreM 2. The reducibility conditions of Theorem 1 are essentially equivalent to the problem of finding two equations of m'* degree differing only in the constant term, each having m distinct integral roots, and the m roots of the first equation all distinct from the m roots of the second equation.

As Dickson has noted,’ this problem came up in the early attempts to find rapidly converging series convenient for the computation of logarithms. A paper by E. B. Escott* contains most of the earlier results and shows how to compute numerical examples for m = 3, --- , 7.

4. Furthermore, it is known’ that the problem of Theorem 2 is equivalent to a special case of the problem of Equal Sums of Like Powers of Diophantine analysis, i.e.,

THEOREM 3. The reducibility conditions of Theorem 1 are essentially equivalent to the problem of finding the distinct integral solutions of the system

6 See Dickson, History of the Theory of Numbers, vol. 2, p. 415. 7L.c., footnote 6, p. 714. 8 Quarterly Journal Math., vol. 41 (1910), pp. 141-167.

CONCERNING CERTAIN REDUCIBLE POLYNOMIALS 73

La= 2 a’ (k= 1,---,m—1).

General solutions of this problem have been found’ for m = 3, 4 and numerous papers” give special solutions and numerical examples for m = 5, --- , 8. These in turn furnish reducible polynomials of the type (3) for n = 8, 10, 12, 14, 16. The following examples involve the smallest primes for degrees 8, 10, 12 that this writer has been able to find.

a(x — 1)(a — 2)(a — 4)(x — 7)(a — 9)(a — 10)(a — 11) +- 179 = [x(x — 4)(x — 7)(x — 11) + 179][(x — 1)(@ — 2)(x — 9)(x — 10) — 179}. (2? — 1)(a? — 5*)(a? — 7?)(a? — 8?)(2? — 9?) + 5039 = [(@ — 1I)@ — 5)(@ + 7)(@ + 8)@ — 9) + 5039) [(z + 1)(@@ + 5)(@ — 7)(x — 8)(@ + 9) — 5039). (x? — 1)(2? — 5*)(2? — 6?)(a? — 9?)(a? — 10*)(2? — 11°) + 100799 = [(z? — 1)(2? — 9?)(2? — 10?) — 100799] [(a? — 5*)(a? — 6?)(2? — 11°) + 100799].

5. Application of method. In conclusion, we shall show that our methods can also be used to determine the integral polynomials of degree n which take + the same integer N 2n times. They are necessarily of the form

(7) a(x — a) --- (tx — a,) + N,

and the conditions to be satisfied are

(8) a(a; — a) --- (aj — an) = +2N (Gi =n+1,---,2n).

By the reasoning previously employed, these are equivalent to

(9) m=hp, o2 = pr, tee, Tn-1 = Pn—1; |on — pn| = 2N/a

or

(10) Ee (kK=1,---,n—1), lon — pn| = 2N/a.

TueoreM 4. Integral polynomials of degree n which take + the same integer N 2n times can exist only for those values of N for which the conditions (9) or (10) can be satisfied.

Examples of these polynomials are:

a(x — 3) + 1 = (# — 1)(@@@ — 2) - 1, a(x — 4)(x — 5) — 6 = (x — 1)(x — 2)(x — 6) + 6, a(x — 4)(x — 7)(x — 11) + 90 = (x — 1)(x — 2)(4 — 9)(x — 10) — 90, ete.

ll

WILLIAMS COLLEGE.

° L. E. Dickson, Introduction to the Theory of Numbers, pp. 49-58. 10 L. ¢., footnote 6, pp. 705-713.

ON SOME CHARACTERIZATIONS OF 2-DIMENSIONAL MANIFOLDS

By Easertus R. vAN KAMPEN

1.1. Object. A large number of papers have been devoted to the problem of finding topological characterizations for 2-sphere, 2-cell or 2-dimensional mani- folds (finite or infinite) of different type. Through complicated cross-citation on the one side, improvements in the available methods on the other side, the study of these papers seems to be at present so much harder than necessary for anybody not already thoroughly acquainted with the ideas used, that the publication of a systematic, simplified exposé of the attained results seems to be the only way of giving these results the place they deserve in the theory of point sets. The results could of course be simplified and extended in several direc- tions. In an appendix we prove that the 2-dimensional generalized manifolds of Cech and Lefschetz are ordinary manifolds.

1.2. Outline of contents and methods. The Theorems I to V” of this paper are very closely related in formulation and proof. This formulation can be reduced to the following scheme. A compact or locally compact Peano space contains at least one curve of one of a few simple types; every curve of that type separates and no closed subset of such a curve separates the space; then the space is homeomorphie with some type of 2-dimensional manifold. In this way we treat in I and III the 2-sphere, in II the closed 2-cell, in IV the 2-dimensional manifold without boundary, in V the open (infinite) 2-dimensional manifolds. The investigation of the set of conditions in IV for a 2-dimensional manifold without boundary was suggested by Zippin. We could have given a characteri- zation of the 2-dimensional manifolds with boundaries by suitably combining the conditions of IIand IV. As the result is less elegant and its formulation and proof do not need any additional idea, we leave this to the reader.

The proofs show of course the effect of the similarity in statement. In later proofs many arguments have been left out simply because they have already occurred before. In the proof of I the most important part is the cutting up of the space by a linear graph in arbitrarily small pieces. Different ways of doing this have been used by Moore, Gawehn, Radé (3). We finally used directly a method of approximating a sum of ares by a linear graph suggested originally by Zippin for the proof of a lemma. The whole argument has been formulated in such a way that it can be used without any change several times more.

Theorems VI and VII contain characterizations by means of Vietoris chains of the compact types of 2-dimensional manifolds. Theorem VII was proved by

Received by the Editors of the Annals of Mathematics February 10, 1934, accepted by them, and later transferred to this journal.

74

Cc

~I

or

CHARACTERIZATIONS OF 2-DIMENSIONAL MANIFOLDS

H. Whitney (14). The theorems are very easily reduced to preceding theorems. We suppose knowledge of the combinatorial notions and theorems used.

In the rest of the paper we give an account of the consequences that can be drawn for Peano spaces from different forms of various parts of the theorem of Janiszewski. Some of the theorems given were proved by Kuratowski (8) and Zippin (9). The proofs in that section are held rather short, but they are all of classical type so that more elaboration seemed superfluous.

1.3. Review of the literature.' In (1) Moore gave three systems of axioms for plane topology, proving in (2) that the spaces determined by his systems were really homeomorphic with the plane. In (3) Moore and Kline announce that certain conditions, of which the important ones are part of or closely related to the Jordan curve theorem, are sufficient to characterize the 2-sphere among point sets situated in 3-space. Their proof does not seem to have been published. In (5) Miss Gawehn proves independently that a slightly modified set of conditions will define 2-dimensional manifolds without boundary among arbitrary Hausdorff spaces.

With (6) the line of attack undergoes a definite change. There Kuratowski announces a system of axioms for spherical topology of which the principal part is formed by the theorem of Janiszewski. His proofs will be found in (8). In the system of axioms given by Woodard in (7) following more closely the scheme of Moore in (1) the most important axiom shows some similarity to the Janiszewski theorem too. In (9) Zippin investigates what kind of Peano continua satisfy the Janiszewski theorem. His results are that in compact (locally compact) locally connected, connected spaces satisfying the Janiszewski theorem, the non- degenerate cyclic elements are homeomorphic with a 2-sphere (a region on a 2-sphere). A systematic account of the results in this line will be found from 9.1 on in this paper.

Again a change takes place. In (10) Wilder announces and in (11) he proves a characterization of the sphere for the purely utilitarian purpose of characterizing the domains determined in 3-space by a sphere. He uses again elements of or related to the Jordan curve theorem. Attacking the problem on its own merits, Zippin then finds in (12) considerably simpler results capable of extension to regions on the sphere. He treats the subject in two different ways (pp. 333-340 and pp. 341-349 of (12)). The results he arrives at by the second treatment cannot be considered as final and are not considered in this paper. The results of the first way of treatment will be found in the theorems I, I’ and V’ of this paper. In (15) Zippin gives a characterization of a closed 2-cell closely analo- gous to the first treatment of the sphere in (12). His result will be found in Theorem II of this paper. In the meantime Roberts (13) had given a char- acterization of 2-dimensional manifolds without boundary and Whitney (14) had given a characterization of the closed 2-cell using an entirely different and

! Numbers in parentheses refer to the list of literature at the end of this paper. Less common terms will be explained in sections 2.1 to 2.4.

76 EGBERTUS R. VAN KAMPEN

very powerful condition (namely the existence of a certain type of chain). See for his result theorem VII of this paper.

2.1. All spaces mentioned in this paper will be separable, metric spaces. We call a space compact, if every infinite number of points in it has a limit point in it; locally compact, if each point has a neighborhood with compact closure; connected, if the space is not the sum of two closed and open subsets that have no point in common; locally connected, if each point is in arbitrarily small connected open sets.

2.2. A locally compact, connected and locally connected space we call P-space (Peano).

A P-space is arewise connected. Any connected open subset of a P-space is again a P-space. If a P-space H is closed in a P-space K, the sum of H and any number of components of K —H is again a P-space.

2.3. A P-space is called cyclic if it has no cut points, that is, if it is not dis- connected by the removal of one point. Any two points on a cyclic P-space K are on a simple closed curve in K; any three points on K are in arbitrary order on an are in K.

If a non-degenerate point set H in a P-space K is not separated on K by any cut point of K,then H determines uniquely a maximal cyclic subset of K, the cyclic element determined by H in K. Such a cyclic element C is a P-space, closed in K, contains all ares in K with end points in C and each point of K — C can be separated from C by a cut point on C.?

2.4. If an are is not determined by a small Greek letter, we may name it by writing down in the correct order all letters representing points on that are that have been named. If @ or pqr is a closed arc, < a > or < pqr > is the cor- responding open arc.

We say that an are spans a point set if it has the two end points and nothing else in common with that point set.

3.1. THEorEMI. A compact P-space D satisfying the following three conditions is homeomorphic with a 2-sphere:

Ia. D contains at least one simple closed curve.

Ib. Every simple closed curve of D separates D.

Ie. No closed arc of a simple closed curve of D separates D2

The proof of this theorem has been so constructed that the last part (4.1) to (4.5) can be used without any change in the proof of later theorems.

3.2. The following condition is equivalent to Ic:

2 We consider here only non-degenerate cyclic elements. For proofs of the theorems mentioned see G. T. Whyburn, On the cyclic connectivity theorem, Bull. Am. Math. Soc., vol. 37 (1931), pp. 429-433. We have substituted the simpler term cyclic for cyclically connected.

3 Ta and Ic could be replaced by the slightly stronger condition: No closed arc of D sepa- rates D. This can be proved by the methods used for Id and If.

CHARACTERIZATIONS OF 2-DIMENSIONAL MANIFOLDS 77

Id. Every component