algebraic topology – application of higher algebra and higher category theory to the study of (stable) homotopy theory
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
In ordinary homology/ordinary cohomology represented as singular homology/singular cohomology, then the cap product operation over a topological space $X$ is the pairing
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
is given on representatives $(\alpha,\beta)$
(where $X/A$ denotes the homotopy cofiber/mapping cone by a map $A \to X$) by the element represented by the composite
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.
Last revised on January 25, 2021 at 15:30:43. See the history of this page for a list of all contributions to it.