nLab Bredon cohomology

Idea

What is called Bredon cohomology after (Bredon 67a, Bredon 67a) is the flavor of ordinary $G$-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 is expressed in (Bredon 67a, p. 3, Bredon 67b, Theorem (2.11), (6.1)), review in in (Greenlees-May, p. 10).

The definition of Bredon cohomology which is more popular (Bredon 67a, p. 2, Bredon 67b, I.6) 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.

Examples

With coefficients in representation ring

Incarnations of rational equivariant K-theory:

	cohomology theory	definition/equivalence due to
$\simeq K_G^0\big(X; \mathbb{C} \big)$	rational equivariant K-theory
$\simeq H^{ev}\Big( \big(\underset{g \in G}{\coprod} X^g\big)/G; \mathbb{C} \Big)$	delocalized equivariant cohomology	Baum-Connes 89, Thm. 1.19
$\simeq H^{ev}_{CR}\Big( \prec \big( X \!\sslash\! G\big);\, \mathbb{C} \Big)$	Chen-Ruan cohomology of global quotient orbifold	Chen-Ruan 00, Sec. 3.1
$\simeq H^{ev}_G\Big( X; \, \big(G/H \mapsto \mathbb{C} \otimes Rep(H)\big) \Big)$	Bredon cohomology with coefficients in representation ring	Ho88 6.5+Ho90 5.5+Mo02 p. 18, Mislin-Valette 03, Thm. 6.1, Szabo-Valentino 07, Sec. 4.2
$\simeq K_G^0\big(X; \mathbb{C} \big)$	rational equivariant K-theory	Lück-Oliver 01, Thm. 5.5, Mislin-Valette 03, Thm. 6.1

Related concepts

• equivariant cohomology, equivariant homotopy theory

• Elmendorf's theorem

• twisted Bredon cohomology

• global equivariant homotopy theory

• global equivariant stable homotopy theory

• orbifold cohomology

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:

• Glen Bredon, Equivariant cohomology theories, Springer Lecture Notes in Mathematics Vol. 34. 1967 (doi:10.1007/BFb0082690)

announced in

• Glen Bredon, Equivariant cohomology theories, Bull. Amer. Math. Soc. Volume 73, Number 2 (1967), 266-268. (educlid:1183528794)

Also:

• Sören Illman, Chapter III in: Equivariant Algebraic Topology, Princeton University 1972 (pdf)

• Sören Illman, Equivariant singular homology and cohomology, Bull. Amer. Math. Soc. Volume 79, Number 1 (1973), 188-192 (euclid:bams/1183534324)

Review:

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

• Tammo tom Dieck, Section II.9 of: Transformation Groups, de Gruyter 1987 (doi:10.1515/9783110858372)

• John Greenlees, Peter May, pages 9-10 of Equivariant stable homotopy theory, chapter 8, pages 277-323 in: Ioan James (ed.), Handbook of Algebraic Topology, North-Holland, Amsterdam, 1995. (doi:10.1016/B978-0-444-81779-2.X5000-7, 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. Gaunce Lewis, Jr., Equivariant Eilenberg-MacLane spaces and the equivariant Seifert-van Kampen suspension theorems, Topology Appl., 48 (1992), no. 1, pp. 25–61 (doi:10.1016/0166-8641(92)90120-O)

Equivalence of Bredon cohomology of topological G-spaces $X$ to abelian sheaf cohomology of the topological quotient space $X/G$ with coefficients a “locally constant sheaf except for dependence on isotropy groups”:

• Hannu Honkasalo, Equivariant Alexander-Spanier cohomology, Mathematica Scandinavia, 63, 179-195, 1988 (doi:10.7146/math.scand.a-12232)

  (for finite groups)

• Hannu Honkasalo, Equivariant Alexander-Spanier cohomology for actions of compact Lie groups, Mathematica Scandinavica Vol. 67, No. 1 (1990), pp. 23-34 (jstor:24492569)

  (for compact Lie groups)

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

  (reformulation in topos theory)

See also at orbifold cohomology.

Equivalent formulation using the Grothendieck construction for a certain Cat-valued presheaf on the orbit category“

• I. Moerdijk, J-A. Svensson, The equivariant Serre spectral sequence, Proc. Amer. Math. Soc. 118 (1993), no. 1, 263–278 doi:10.2307/2160037

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

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

• Amiya Mukherjee, Goutam Mukherjee, Bredon-Illman cohomology with local coefficients, The Quarterly Journal of Mathematics, Volume 47, Issue 2, June 1996, Pages 199–219 (doi:10.1093/qmath/47.2.199)

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

• Samik Basua, Debasis Sen, Representing Bredon cohomology with local coefficients, Journal of Pure and Applied Algebra Volume 219, Issue 9, September 2015, Pages 3992-4015 (doi:10.1016/j.jpaa.2015.02.001)

On cases where $\mathbb{Z}/p$-equivariant Bredon cohomology groups are free modules over the Bredon cohomology of the point:

• Kevin K. Ferland, L. Gaunce Lewis, Jr., The $RO(G)$-Graded Equivariant Ordinary Homology of $G$-Cell Complexes with Even-Dimensional Cells for $G=\mathbb{Z}/p$, Memoirs of the AMS 167 (2004) [ams:memo-167-794, 978-ISBN:1-4704-0392-8]

Specifically on Kronholm's freeness theorem for $\mathbb{Z}/2$-equivariant Bredon cohomology:

• William C. Kronholm, A Freeness Theorem for $RO(\mathbb{Z}/2)$-graded Cohomology, Topology and its Applications 157 5 (2010) 902-915 [arXiv:0908.3825, doi:10.1016/j.topol.2009.12.006]

• Eric Hogle, Clover May, The freeness theorem for equivariant cohomology of $Rep(C_2)$-complexes, Topology and its Applications 285 1 (2020) 107413 [arXiv:2005.07300, doi:10.1016/j.topol.2020.107413]

• Daniel Dugger, Christy Hazel, Clover May, Equivariant $\mathbb{Z}/\ell$-modules for the cyclic group $C_2$, Journal of Pure and Applied Algebra 228 3 (2024) 107473 [arXiv:2203.05287, doi:10.1016/j.jpaa.2023.107473]