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slant product

In algebraic topology, the slant product in homology 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 AA has a commutative ring structure, one can take A=AA'=A and postcompose with AAAA\otimes A\to A 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=*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.

Likewise, the slant product in cohomology is a map of the form

H i(Y,A)H n(X×Y,A)H ni(X,AA).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.