nLab cap product



Algebraic topology



Special and general types

Special notions


Extra structure





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. 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)


Last revised on January 25, 2021 at 15:30:43. See the history of this page for a list of all contributions to it.