# 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 [Dold], VII.11 and VII.13.

\bibitem{Dold} Albrecht Dold, Lectures on Algebraic Topology, Springer, 1980.

Last revised on January 29, 2019 at 20:37:22. See the history of this page for a list of all contributions to it.