# nLab slant product

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

${H}_{q}\left(X,A\right)\otimes {H}^{n}\left(X×Y,A\prime \right)\to {H}^{n-q}\left(Y,A\otimes A\prime \right).$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

$\mathrm{Chains}\left(X\right)\otimes \mathrm{Chains}\left(Y\right)\to \mathrm{Chains}\left(X×Y\right).$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

${H}_{q}\left(X,A\right)\otimes {H}^{n}\left(X×Y,A\right)\to {H}^{n-q}\left(Y,A\right).$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

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