nLab cap product

Idea

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

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

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 $E$, then the cap product

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

is given on representatives $(\alpha,\beta)$

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

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

$$S^{p+q} \overset{\beta}{\longrightarrow} E \wedge X/A \overset{id \wedge \Delta}{\longrightarrow} E \wedge X/A \wedge X \overset{id \wedge \alpha \wedge id}{\longrightarrow} E \wedge \Sigma^p E \wedge X \overset{\mu \wedge id}{\longrightarrow} \Sigma^p E \wedge X$$

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

Related concepts

• cup product

• intersection pairing

• Poincaré duality

• fundamental class

References

• Stanley Kochman, (4.2.1) and prop. 4.2.10 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996