nLab
slant product

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

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

$$\mathrm{Chains}(X)\otimes \mathrm{Chains}(Y)\to \mathrm{Chains}(X\times Y).$$Chains(X)\otimes Chains(Y)\to Chains(X\times Y).

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

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

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