nLab
slant product

In algebraic topology, the slant product is the following pairing between singular homology and singular cohomology:

H q(X,A)H n(X×Y,A)H nq(Y,AA). H_q(X,A)\otimes H^n(X\times Y,A')\to H^{n-q}(Y,A\otimes A').

It is induced at the chains/cochains level by the Eilenberg-Zilber chain map

Chains(X)Chains(Y)Chains(X×Y). Chains(X)\otimes Chains(Y)\to Chains(X\times Y).

When the abelian group AA has a commutative ring structure, one can take A=AA'=A and postcompone with AAAA\otimes A\to A to obtain the pairing

H q(X,A)H n(X×Y,A)H nq(Y,A). H_q(X,A)\otimes H^n(X\times Y,A)\to H^{n-q}(Y,A).

In particular, for Y=*Y=* one obtains the contraction

H q(X,A)H n(X,A)H nq(*,A) H_q(X,A)\otimes H^n(X,A)\to H^{n-q}(*,A)

taking values in the coefficient ring of the given cohomology theory.

Revised on April 26, 2010 18:07:20 by Urs Schreiber (131.211.232.147)