slant product

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

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

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

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

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

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

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

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

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)