Annals of Mathematics

On Duality and Intersection Chains in Combinatorial Analysis SitusAuthor(s): J. H. C. WhiteheadSource: Annals of Mathematics, Second Series, Vol. 33, No. 3 (Jul., 1932), pp. 521-524Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1968532 .

Accessed: 15/11/2014 00:33

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Annals ofMathematics.

http://www.jstor.org

This content downloaded from 129.20.73.3 on Sat, 15 Nov 2014 00:33:16 AMAll use subject to JSTOR Terms and Conditions

ON DUALITY AND INTERSECTION CHAINS IN COMBINATORIAL ANALYSIS SITUS.*

By J. I{. C. WHITEHEAD.

1. Introductory. This note is intended as a contribution to the formal exposition of combinatorial analysis situs. We do not attempt to cover much ground, but are content with a proof of the fundamental relation

F(Cp *C) - Cp * F( W7) + (- l)1 F(C1,) - C,

which is due to S. Lefschetz.' An account of the geometry underlying this note is to be found in his Colloquium Lectures.2 The purely com- binatorial theory has been developed in a series of articles by J. W.Alexander and by M. H. A. Newman,3 further references to which are made in "Topo- logy". We also call attention to a paper by Newman' and to Chap. 11 of a memoire by G. de Rahm,5 in which intersection chains are defined by means of looping coefficients without reference to the dual complex.

2. Derived chains. Let M1n be an orientable and oriented manifold

(2.1) ( +1)l ? Ca.ao nX% x... H(i understood for a = 0, 1,... N), 21) (n+ 1)! where the coefficients c are -1, 0 or + 1, and are anti-symmetric in their indices. The symbols x obey the multiplication law xx' -x' x. All relations will be relative, in the sense of Lefschetz, to F(Ma), the boundary of JIR,,. Formally this is the same as assuming F(ilh) = 0. For non- orientable manifolds take everything mod. 2.

The first derived complex D (M.) is composed of cells

(2.2) _ Mao Xaoa, * * a, *

where xc0. .. is the new vertex6 associated with the p-cell xac< Xaps

It follows by an easy induction that the cell (2.2), with the + sign, is similarly oriented to the cell xa. xa. Therefore

* Received December 8, 1931. I Trans. American Math. Soc. 28 (1926), pp. 1-49. 2 Lefschetz, Topology, New York, 1930. See, in particular, the beginning of Chapters II]

and IV. I See for instance, Alexander, Annals of Math. 31 (1930), pp. 292-320; and Newman

Proc. Royal Academy of Amsterdam, 29 (1926), pp. 611 and 627. 4Proc. Cambridge Phil. Soc. 27 (1931), pp. 497-507. 5 Journal de Math. p. et a. 10 (1931), pp. 115-200. 6 This convenient notation was, I believe, introduced by Alexander.

521

This content downloaded from 129.20.73.3 on Sat, 15 Nov 2014 00:33:16 AMAll use subject to JSTOR Terms and Conditions

522 J. H. C. WHITEHEAD.

(2.3) D (iMl) = C *Xa0 Xx0a, . . .XaO .an

where the c's are the same as those in (2.1). Let

CP (r + 1)! XtXa

be any chain, the t's being positive, negative, or zero, and anti-symmetric in their indices. Its boundary is given by

F(C) .. *a

where f, like all Greek indices, runs from 0 to N. A useful formula follows from the fact that the boundary of D(Cp) is the derived complex of F(Cp). That is to say

(2.4) FD(Cp) D F (Cp) = l 'Xa Xaa2 a..a

3. Dual chains. The complement of an r-cell in M.,

Er = o Xa *ar

is given by

(3.1) (n- r) C XA * X., x (q = n-r)

and the chain dual to Er by

(3.2) E * = ZSC Al A 18A2 ... gAl where

(3.3) { = AO a

The first derived chain of (3.1) is given by

Sq-1 cao r. Al .. tt ... X) ... 2

and by the definition of a manifold is a sphere, rel. F(Mn) (whether a "true" or a "combinatory" sphere is irrelevant to this discussion). The congruence x -> carries xSq-i into Eq and therefore the latter is a cell.

An orientation for any two of the chains M,, Er and Eq* determines an orientation for the third. For if a0 ... ar ... )q are indices such that

Cao * ar 21 * A2{ + 0.

This content downloaded from 129.20.73.3 on Sat, 15 Nov 2014 00:33:16 AMAll use subject to JSTOR Terms and Conditions

ON DUALITY AND INTERSECTION CHAINS. 523

these three chains may be oriented by assigning a sense to the orders ao *r 21 * q ao ar and 2 **q respectively. An order for 21 * * determines an orientation for F(E), and (3.2) fixes the position of the vertex7 d.

4. The intersection of chains. If

Ep -~ t: p is any cell in -31L, then8

(4.1) D(Ep) - .p XAo X,...P

For (4.1) and (3.2) to have a vertex in common it is necessary and sufficient that the a's are contained among the Xl's. That is to say p > r and Er must be incident with Ep. For simplicity of notation let

Er -Xo Xl Xr

Ep Xo ... XrX+l ... Xp

Then the intersection Epic Eq* is given by

(4.2) Ep- Eq 0

where s = p - r, the i's are given by (3.3), and

0A1 ..A" = A -2 r+'1"p.

It follows by arguments similar to those used in ? 3 that Ep~ Eq is a cell, and that an orientation for any three of Min Ep, Eq and Epc Eq* determines an orientation for the fourth. This is in accordance with the conditions stated in Chap. IV of "Topology". We suppose the cells in question to be positively oriented as written. From (4.2) and (2.4) we have

(4-3) F (Ep Eq*) 0i1 A'tA .* .i . A2 ...f 2 A ** 2- A

I This formalism should be convenient for a discussion of the Poincar6 duality relations. For example, the relation

D (M.) - D (M.*)

is evident when we set r - 0 in (3.2), replace a by x according to (3.3), and sum over a*. We then have

D (.M*) - To Ea c ? ...X h --

which is identical with (2.3). 8The "generalized Kronecker d" is 0 unless )o ... AP are obtained by permuting

B, *...*, ,p, and is + 1 for even, - 1 for odd permutations.

This content downloaded from 129.20.73.3 on Sat, 15 Nov 2014 00:33:16 AMAll use subject to JSTOR Terms and Conditions

524 J. H. C. WHITEHEAD.

while from (3.2) we have

F(Eq*) 2- 1..rAi *.Aq

since a Eq1l t is a sphere and its boundary vanishes. Therefore

(4.4) EP * F(Eq ) - 'AA

We also have, from (2.4)

F(E,) - d, .'2.. *X. .,

and F (E).E* (1-FI )r4

1 0 r62A2 A;

* ( )r-+l t s vA~~~~~

..............r......... p 22 , 2 .... 2, (4.5)

p q ~ r pA

the factor (-1)r+? being present because o, as an index on d, has been shifted r + 1 places to the right. From (4.3), (4.4) and (4:5) we have

F(Ep , E ) Ep*F(E,*)+(-1) F (Ep) - E

which leads to the fundamental relation for chains

(46 Fx (C - C*) S

p-F C*, .

)n-qF(C ,, - ,*.

This content downloaded from 129.20.73.3 on Sat, 15 Nov 2014 00:33:16 AMAll use subject to JSTOR Terms and Conditions