Modular Homology Characters

VOL. 18, 1932

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1 The notation £(of), which may be read: "geometric a," is introduced in order to keep clear the distinction between a physical magnitude, such as a = 17 ounces, and the geometric length which is used to represent this magnitude on the diagram. By the aid of this notation it is possible to adhere strictly to the useful couvention that "the equality sign should never be used except between quantities of the same kind." Thus, if a = 17 ounces, we may have £(a) = 2.1 inches on the diagram; but we could not properly write a = 2.1 inches, since 2.1 inches is not equal to 17 ounces. [If, however, u is itself a geometric length on the diagram, then £(u) = u. ] 2 The mathematical proof of the correctness of this construction, together with a more extensive table of the factor K, will be given in a forthcoming paper in the Journal of the American Statistical Association, September, 1932.

MOD ULAR HOMOLOGY CHARA CTERS BY A. W. TUCKER DEPARTMENT OF MATHEMATICS, PRINCETON UNIVERSITY

Communicated May 9, 1932

This note discusses the theory of modular homology characters founded by Alexander in his paper on Combinatorial Analysis Situs1 (hereafter designated by A. C.), and more recently considered by Lefschetz in his Colloquium Lectures on Topology2 (hereafter designated by L. T.). We preface our work by an elementary treatment of matrices equivalent modulo m (m being any integer > 1 or, suitably interpreted, = 0); this treatment is essentially related to the methods used by Alexander in handling modular chains (cf. pp. 319-22 A. C.). 1. Matrices Equivalent Mod m.-Let w denote a matrix the elements of which are integers w2,. We say that another such matrix 'w is equivalen mod m to w if ,Wd xd u,, y mod m, (1.1) where x and y are square matrices of integers with determinants lxl and IYI prime to m. This equivalence is clearly reflexive and transitive. It is also symmetric, for Uw -X-d w5 yy mod m, (1.2)

wlxere 'x is a matrix obtained by taking 'xd = times the cofactor of xcd in x, t being such that [xl. u 1 mod m, and 'y is a matrix obtained in similar fashion from y. If m = 0 our equivalence mod m becomes ordinary equivalence; we have only to interpret congruence mod 0 as equality and agree that i 1 - which are the only integral solutions of xl. = 1 are the only integers prime to 0.

MATHEMATICS: A. W. TUCKER

468

PROC. N. A. S.

Let G(m) (w) denote the G. C. D. of the values of all the polynomials,

P,('(w)

=

aomT + ai(w)m'-1 +

...

+

ar.1(w)m + a7(w),

which can be formed by taking for ao an arbitrary integer and for ap(w) a form with arbitrary integral coefficients linear in the p-rowed determinants Ap(w) of w. (We agree that the G. C. D. of a set of integers is that of the non-zero members, if anv; otherwise it is zero.) The number of Gtm) (w)'s, excluding those (if any) for which G(m) (w) _ 0 mod m.Gr(1) (w), we call the rank mod m of w and denote by p(m) (w). Clearly G(°) (w) is the G. C. D. of all Ar4(w), and p(0)(w) the ordinary rank of w. By expressing (1.1) as = + mk, w^, =

xwdy,c

we have, using familiar properties of determinants, that a Ap('w) is a pP(m) (W) and that a A. (w) is an a,(w). Accordingly, a pr: ) ('w) is a p(m) (w). In similar fashion it follows from (1.2) that a P(m)(w) is a Plm) ('w). Therefore

G(m) ('w)

G(m) (w),

(1.3)

and so

P(m)(1w)

= P(m) (W)

(1.4)

Compatible with our definition of equivalence mod m are the four following elementary transformations mod m (cf. the "generating transformations," p. 320 A. C.): A. The addition of the elements of one row (column) to the corresponding elements of another row (column). B. The reversal of the signs of all the elements of a row (column). C. The multiplication of all the elements of a row (column) by one and the same positive integer pnme to m. D. The replacement of any element by a congruent integer mod m. If m = 0, C and D become identity transformations. Clearly the inverses of all four transformations may be achieved by combinations of elementary transformations. THEOREM. By elementary transformations mod m the matrix w may be reduced to a unique matrix v consisting of zeros except down the main diagonal where there are elements I(^)(w) (s = 1, 2, ... , p(M)(w)) such that

I(`)(w)

=

G(")(w)/G,(!)i(w) (Go")(w)

=

1).

(1.5)

By transformations of types A and B, w can be reduced to the well-known normal form consisting of zeros except for the ordinary invariant factors I(°)(w) down the main diagonal. Now using transformations of types C and D, if m 5 0, passage is made to a matrix v in which each I(')(w) is

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replaced by the G. C. D. of it and m, or by 0 if it is divisible by m. The number of these non-zero elements is clearly p() (v), i.e., p(m) (w) in virtue of (1.4). We call them the invariant factors mod m of w, and denote them by (M) (w). Since by construction each I,() (w) divides its successors, if any, calculation shows that G(m) (v) - 1(m)(w).I(m)(w) (m)"(w), from which (1.5) follows in consequence of (1.3). Taken together (1.3) and (1.5) imply that IgM) (1W) IS() (W), thereby showing that the reduced matrix v is unique. The above theorem has as corollary the fact that the passage from w to any matrix equivalent mod m, say 'w, can be made by elementary transformations mod m; we have only to reduce w to v by elementary transformations and then reverse those reducing 'w to v. We call the powers of primes occurring in the invariant factors (M) (w) the elementary divisors mod m of w. It is readily seen that the elementary divisors mod m of a matrix, whose only elements not = 0 mod m lie (anyhow) on a diagonal, are just the highest powers of primes these elements have in common with m. 2. Homology Characters Mod m.-Let ?7" be the matrix exhibiting the incidences between the (p + l)-cells and the p-cells of an n-complex K. Let

p(o) W"§)

=

IAO)('r)

=

p

,t

( ( p1)pi) Io,") (rP) = el,

Let {Cp} (i = 1, 2, . . .., oap; ap = the number of p-cells of K) denote a set of p-chains on K whose coefficients form a unimodular matrix. We can choose3 a basis Cp }, for each value of p, such that

I

CP,+1

0°

(K =1, 2*.

ap+1-pp)) >

(2.1)

r)1PCp" (A = lp 20 ..* .}Pp) The group rPm)of p-cycles mod m will be generated by cp+I+lCP+1

{Ct, {1Cp}, {GCIP }, (fi =1,2, a

..,PP; v=PP + 1,**.,°sP-1; ape- PP-I1 + 1..*,ap),

where The members of (2.2) are subject to the homologies:

(2.2)

4470

MATHEMATICS: A. W. TUCKER

mC Cp

0

modm.

PRoc. N. A. S.

(2.3)

O 1-v)Cp tap c+1

Let FP"') denote the sub-group of rp) generated by the left members of (2.3), and let cop denote the diagonal matrix of the coefficients appearing in (2.3). The difference (factor) group t(m) = rim) - Fpm) is the pth homology group mod m of K. Let

Rp(K, m) cNp Pp- Pp-is Rp(K, m) = ap - p These two numbers are both numerical invariants of K; they are candidates for the position of pth Betti number mod m of K. The formula for Rp(K, m) is an obvious generalization of the formula for Rp(K), but on the other hand Rp(K, m) is the maximum number of cycles yp mod m between which there holds no relation

tj,y1 + t2,y'

+

.

mod m,

unless all t = 0 mod m. (If the latter statement were not true the invariant factors of cop would not be unique.) Rp(K, m) is the modular Betti number defined on p. 42 L. T., while Rp(K, m) is that implicit in the remarks on p. 35 L. T.4 The former yields an Euler-Poincard formula, but not the latter. For 7r prime

Rp(K, 7ru) = Rp(K, 7ru) (2.4) This accounts for the single definition of Betti (connectivity) number mod 7r on p. 319 A. C. Let et(K, m) denote the invariant factorsI) (cop) > 1. These numerical invariants of K are the torsion coefficients mod m defined on p. 43 L. T. They completely determine the group kp ). The reduction of wp to normal form is the counterpart of the resolution of M4(p) into thSe direct sum of cyclic groups of "nested" orders eO(K, m). On the other hand, let o,1(K, m) denote the invariant factors Ik() (w) > 1. Since all the diagonal elements of cop divide m the 5,'P(K, m) merely differ from e,(K, m) by the loss of RY(K, m) m's. Hence Rp(K, m) and the 5(K, m) are numerical invariants characterizing Hp"). More important, however, than the characterization of Ip(m) by the invariant factors of wp is the characterization by its elementary divisors. This latter corresponds to the resolution of Hp() into a direct sum of cyclic groups having orders which are powers of primes. THEOREM I. i4 ' is characterized by the number Rp(K, m) and the elementary divisors mod m of rP and n-. The proof is immediate, for in consequence of our remark at the end of § 1 we have only to read down

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(2.3) to see that the elementary divisors of cop are: (1) the elementary divisors mod m of qtp, (2) the powers of primes composing m repeated Rp(K, m) times, and (3) the elementary divisors mod m of VP'-. Since a passage from the matrices tp to equivalent matrices mod m does not affect the ranks p4 or the elementary divisors mod m, Theorem I shows that the homology groups Hp() are also unchanged. This means that for considerations mod m it is sufficient to know incidences to within a congruence mod m. Let K have a dual K*. Then THEOREM II. The homology groups Hpm) and H*"n' of K and K*, respectively, are isomorphic.5 The incidence matrix ti*n P- lof K* is the same as the transverse (transpose) of tP except perhaps for a uniform difference of sign. Hence the elementary divisors mod m of 1 n- p-1 are those of 7t7 and of n"',S those of ti"1. Moreover,

P(K*, m)

Rn

=

a*..p-Pr-p-Pu-p-i

= a-p

-i

- p =

Rp(K, i).

Consequently the proof of H is completed by applying I. THEOREM III. The numbers Rp(K, m)for all m which are powers of primes, and for all p, determine all Rp(K) and all Op(K).6 Let m = 7rU and let tpU) = pp- pp,. Then t'U- 1) - tpU) = the number of times 7rU is an elementary divisor of nt. To calculate each t(U) we have the formula

Rp(K, 7ru)

Rp(K) = t(u) + t4u) = 0 = pn. But tpU) =

-

0, all p, for u sufwhere t(T) = 0 = t'u) since P-i ficiently large and so Rp(K) can be calculated. Hence a knowledge of Rp(K, ru), all u and all p, determines Rp(K) and the elementary divisors of -ti which are powers of ir, all p. By applying this result to all primes 7r the theorem is obtained. Due to (2.4) Rp(K, m) might have replaced Rp(K, m) in the statement of the theorem. 1 J. W. Alexander, Trans. Am. Math. Soc., 28, 301-329 (1926).

S. Lefschetz, Topology, New York (1930). The topological terminology and notation we use are based on this book. 3 The choice of bases is like that on p. 37 L. T. except that the elements have been permuted so that the reduced matrices have the invariant factors arranged in ascending order up the diagonal originating at the lower left corner. 4 Professor Alexander in his seminar (April, 1931) discussed the difference between these definitions, and showed that both led to numerical invariants. The matter had been brought to light by an example constructed by Hausdorff and communicated to Alexandroff at Princeton. (ech has recently given an example also. 6 Cf. L. Pontrjagin, Math. Annalen, 105, 169 (1931). 6 Cf. pp. 322-323 A. C. The phrase "greater than or equal to ir" occurring in the second last line of p. 322, and again in the next sentence, should of course read "divisible by xr." 2