nLab
Borel equivariant cohomology

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Representation theory

Contents

Idea

For XX a space and GG a group acting on it, then the Borel equivariant cohomology of XX is the cohomology of the homotopy quotient X//GX//G.

Since a standard way to model the homotopy quotient is the Borel construction, this is called Borel equivariant cohomology.

This is the special case of genuine equivariant cohomology where the action on the coefficients is trivial.

Properties

As right induced genuine equivariant cohomology

In the more general context of (global) equivariant stable homotopy theory, Borel-equivariant spectra are those which are right induced from plain spectra, hence which are in the essential image of the right adjoint to the forgetful functor from equivariant spectra to plain spectra.

(Schwede 18, Example 4.5.19)

Examples

cohomology in the presence of ∞-group GG ∞-action:

Borel equivariant cohomologyAAAAAA\phantom{AAA}\leftarrow\phantom{AAA}general (Bredon) equivariant cohomologyAAAAAA\phantom{AAA}\rightarrow\phantom{AAA}non-equivariant cohomology with homotopy fixed point coefficients
AAH(X G,A)AA\phantom{AA}\mathbf{H}(X_G, A)\phantom{AA}trivial action on coefficients AAAA[X,A] GAA\phantom{AA}[X,A]^G\phantom{AA}trivial action on domain space XXAAH(X,A G)AA\phantom{AA}\mathbf{H}(X, A^G)\phantom{AA}

References

Last revised on June 27, 2019 at 09:16:30. See the history of this page for a list of all contributions to it.