nLab
cap product

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

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Contents

Idea

In ordinary homology/ordinary cohomology represented as singular homology/singular cohomology, then the cap product operation over a topological space XX is the pairing

H p(X)H p+q(X)H q(X) H^p(X)\otimes H_{p+q}(X) \longrightarrow H_q(X)

given by combining the Kronecker pairing of the cohomology class with the image of the homology class under diagonal and using the Eilenberg-Zilber theorem.

More generally, in relative generalized (Eilenberg-Steenrod) cohomology/generalized homology represented by a ring spectrum EE, then the cap product

E p(X,A)E (p+q)(X,A)E q(X) E^p(X,A) \otimes E_{(p+q)}(X, A) \longrightarrow E_q(X)

is given on representatives (α,β)(\alpha,\beta)

[X/AαΣ pE],[S p+qβEX/A] [X/A \overset{\alpha}{\longrightarrow} \Sigma^{p} E] \,, \;\;\; [S^{p+q} \overset{\beta}{\to} E \wedge X/A]

(where X/AX/A denotes the homotopy cofiber/mapping cone by a map AXA \to X) by the element represented by the composite

S p+qβEX/AidΔEX/AXidαidEΣ pEXμidΣ pEX S^{p+q} \overset{\beta}{\longrightarrow} E \wedge X/A \overset{id \wedge \Delta}{\longrightarrow} E \wedge X/A \wedge X \overset{id \wedge \alpha \wedge id}{\longrightarrow} E \wedge \Sigma^p E \wedge X \overset{\mu \wedge id}{\longrightarrow} \Sigma^p E \wedge X

(e.g. Kochmann 96, p. 136)

This pairing induces a corresponding pairing on Atiyah-Hirzebruch spectral sequences (in the manner discussed at multiplicative cohomology theory/multiplicative spectral sequence), such that on the second page it restricts to the cap prodct in ordinary cohomology.

(e.g. Kochmann 96, p. 138)

References

Last revised on April 27, 2017 at 16:20:58. See the history of this page for a list of all contributions to it.