group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
By real cohomology one usually means ordinary cohomology with real number coefficients, denoted $H^\bullet\big(-, \mathbb{R}\big)$.
Hence, with the pertinent conditions on the domain space $X$ satisfied, its real cohomology $H^\bullet\big(-, \mathbb{R}\big)$ is what is computed by the Cech cohomology or singular cohomology or sheaf cohomology of $X$ with coefficients in $\mathbb{R}$.
In particuar, for $X$ a smooth manifold, the de Rham theorem says that real cohomology of $X$ is also computed by the de Rham cohomology of $X$
More generally, for $X$ a smooth manifold with smooth action of a connected compact Lie group, the equivariant de Rham theorem says that the real cohomology of the homotopy quotient (e.g. Borel construction) of $X$ is computed by the Cartan model for equivariant de Rham cohomology on $X$.
cohomology | Borel-equivariant cohomology |
---|---|
real ordinary cohomology | real equivariant ordinary cohomology |
de Rham cohomology | equivariant de Rham cohomology |
Last revised on December 5, 2020 at 10:50:25. See the history of this page for a list of all contributions to it.