nLab slant product

In algebraic topology, the slant product in homology 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 postcompose 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.

Likewise, the slant product in cohomology is a map of the form

$$H^i(Y,A)\otimes H_n(X\times Y,A')\to H_{n-i}(X,A\otimes A').$$

It is induced by the Alexander-Whitney map.

There are also versions for relative homology and relative cohomology. See Dold, VII.11 and VII.13.

• Albrecht Dold, Lectures on Algebraic Topology, Springer, 1980.