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
Hochschild cohomology, cyclic 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 10:30:43. See the history of this page for a list of all contributions to it.