slant product

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

H q(X,A)H n(X×Y,A)H nq(Y,AA).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)Chains(Y)Chains(X×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=A and postcompone with AAA to obtain the pairing

H q(X,A)H n(X×Y,A)H nq(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)H n(X,A)H nq(*,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.

Revised on April 26, 2010 18:07:20 by Urs Schreiber (