slant product

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

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

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

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

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

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

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)