# nLab Bredon cohomology

Contents

cohomology

### Theorems

#### $(\infty,1)$-Topos Theory

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

# Contents

## Idea

What is called Bredon cohomology after (Bredon 67) is the flavor of $G$-equivariant cohomology which uses the “fine” equivariant homotopy theory of topological G-spaces that by Elmendorf's theorem is equivalent to the homotopy theory of (∞,1)-presheaves over $G$-orbit category, instead of the “coarse” Borel homotopy theory. See at Equivariant cohomology – Idea for more motivation.

For more technical details see there equivariant cohomology – Bredon equivariant cohomology.

## Definition

Let $G$ be a compact Lie group, write $Orb_G$ for its orbit category and write $PSh_\infty(Orb_G)$ for the (∞,1)-category of (∞,1)-presheaves over $Orb_G$. By Elmendorf's theorem this is equivalent to the homotopy theory of topological G-spaces with weak equivalences the $H$-fixed point-wise weak homotopy equivalences for all closed subgroups $H$ (“the equivariant homotopy theory”):

$\mathbf{H}^{Orb_G^{op}} \coloneqq L_{fpwe} G Top \simeq PSh_\infty(Orb_G) \,.$

A spectrum object $E \in Stab(\mathbf{H}^{Orb_G^{op}})$ in the (∞,1)-topos $\mathbf{H}^{Orb_G^{op}}$ is what is called a spectrum with G-action or, for better or worse, a “naive G-spectrum”.

For $X$ a G-space, then its cohomology in $\mathbf{H}^{Orb_G^{op}}$ with coefficients in such $A$ might be called generalized Bredon cohomology (in the “generalized” sense of generalized (Eilenberg-Steenrod) cohomology).

Specifically for $n \in \mathbb{N}$ and $A \in Ab(Sh(Orb_G))$ an abelian sheaf then there is an Eilenberg-MacLane object

$K(n,A) \in \mathbf{H}^{Orb_G^{op}}$

whose categorical homotopy groups are concentrated in degree $n$ on $A$.

Then ordinary Bredon cohomology (in the “ordinary” sense of ordinary cohomology) in degree $n$ with coefficients in $A$ is cohomology in $\mathbf{H}^{Orb_G^{op}}$ with coefficients in $K(n,A)$:

$H_G^n(X,A) \simeq \pi_0 \mathbf{H}^{Orb_G^{op}}(X,A)$

(see the general discussion at cohomology).

If here $X$ is presented by a G-CW complex and hence is cofibrant in the model category structure that presents the equivariant homotopy theory (see at Elmendorf's theorem for details), then the derived hom space on the right above is equivalently given by the ordinary $G$-fixed points of the ordinary mapping space of the topological space underlying the G-spaces.

$H_G^n(X,A) \simeq \pi_0 [X,A]_G \,.$

In this form ordinary Bredon cohomology appears for instance in (Greenlees-May, p. 10).

But what (Bredon 67) really wrote down is a chain complex-model for this: regarding $X$ again as a presheaf on the orbit category, define a presheaf of chain complexes

$C_\bullet(X) \;\colon\; Orb_G^{op}\longrightarrow Ch_\bullet$

by

$C_n(X)(G/H) \coloneqq H_n((X^n)^H, (X^{n-1})^H, \mathbb{Z}) \,,$

where on the right we have the relative homology of the CW complex decomposition underlying the G-CW complex $X$ in degrees as indicated. The differential on these chain complexes is defined in the obvious way (…).

Then one has an expression for ordinary Bredon cohomology similar to that of singular cohomology as follows:

$H_G^n(X,A) \simeq H_n(Hom_{Orb_G}(C_\bullet(X,)A)) \,.$

(due to Bredon 67, see e.g. (Greenlees-May, p. 9)).

More generally there is $RO(G)$-graded equivariant cohomology with coefficients in genuine G-spectra. This is also sometimes still referred to as “Bredon cohomology”. For more on this see at equivariant cohomology – Bredon cohonology.

cohomology in the presence of ∞-group $G$ ∞-action:

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

## References

The original text is

announced in

Reviews include

• Andrew Blumberg, section 1.4 of Equivariant homotopy theory, 2017 (pdf, GitHub)

• John Greenlees, Peter May, pages 9-10 of Equivariant stable homotopy theory (pdf)

• Peter May, section I.4 of Equivariant homotopy and cohomology theory CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996. With contributions by M. Cole, G. Comezana, S. Costenoble, A. D. Elmenddorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. (pdf, pdf)

• Paolo Masulli, section 2 of Equivariant homotopy: $KR$-theory, Master thesis (2011) (pdf)

The Eilenberg-MacLane objects over the orbit category are discussed in detail in

• L. G. Lewis, Equivariant Eilenberg-MacLane spaces and the equivariant Seifert-van Kampen suspension theorems, Topology Appl., 48 (1992), no. 1, pp. 25–61.

There is an interesting (nontrivially) equivalent definition by Moerdijk and Svensson, using the Grothendieck construction for a certain $Cat$-valued presheaf on the orbit category.

Further remarks on this and on the twisted cohomology-version is in

• G. Mukherjee, N. Pandey, Equivariant cohomology with local coefficients (pdf)

• H. Honkasalo, Sheaves on fixed point sets and equivariant cohomology, Math. Scand. 78 (1996), 37–55 (pdf)

• H. Honkasalo, A sheaf-theoretic approach to the equivariant Serre spectral sequence, J. Math. Sci. Univ. Tokyo 4 (1997), 53–65 (pdf)

For orbifolds there is a generalization of $K$-theory which is closely related to the Bredon cohomology (rather than usual equivariant cohomology):

Last revised on May 19, 2019 at 13:21:42. See the history of this page for a list of all contributions to it.