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Bredon cohomology

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

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

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

Idea

What is called Bredon cohomology after (Bredon 67a, Bredon 67a) is the flavor of ordinary GG-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 GG-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 GG be a compact Lie group, write Orb GOrb_G for its orbit category and write PSh (Orb G)PSh_\infty(Orb_G) for the (∞,1)-category of (∞,1)-presheaves over Orb GOrb_G. By Elmendorf's theorem this is equivalent to the homotopy theory of topological G-spaces with weak equivalences the HH-fixed point-wise weak homotopy equivalences for all closed subgroups HH (“the equivariant homotopy theory”):

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

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

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

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

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

whose categorical homotopy groups are concentrated in degree nn on AA.

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

H G n(X,A)π 0H Orb G op(X,A) H_G^n(X,A) \simeq \pi_0 \mathbf{H}^{Orb_G^{op}}(X,A)

(see the general discussion at cohomology).

If here XX 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 GG-fixed points of the ordinary mapping space of the topological space underlying the G-spaces.

H G n(X,A)π 0[X,A] G. 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 XX again as a presheaf on the orbit category, define a presheaf of chain complexes

C (X):Orb G opCh C_\bullet(X) \;\colon\; Orb_G^{op}\longrightarrow Ch_\bullet

by

C n(X)(G/H)H n((X n) H,(X n1) H,), 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 XX 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)H n(Hom Orb G(C (X,)A)). 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)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 theorydefinition/equivalence due to
H ev((hHX h)/G;) \phantom{\simeq} H^{ev}\Big( \big(\underset{h \in H}{\coprod} X^h\big)/G; \mathbb{C} \Big) delocalized equivariant cohomologyConnes-Baum 89
H CR ev((XG);)\simeq H^{ev}_{CR}\Big( \prec \big( X \!\sslash\! G\big);\, \mathbb{C} \Big)Chen-Ruan cohomology
of global quotient orbifold
Chen-Ruan 00
H G ev(X;(G/HRep(H)))\simeq H^{ev}_G\Big( X; \, \big(G/H \mapsto \mathbb{C} \otimes Rep(H)\big) \Big)Bredon cohomology
with coefficients in representation ring
Honkasalu 90, Thm. 5.5
Moerdijk 02, p. 18
K G 0(X;)\simeq K_G^0\big(X; \mathbb{C} \big) rational equivariant K-theoryLück-Oliver 01, Thm. 5.5

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

The original text is

announced in

Also

Reviews include

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

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

See also at orbifold cohomology.

Equivalent formulation 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

  • 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)

See also at orbifold cohomology.

Last revised on October 19, 2020 at 04:06:02. See the history of this page for a list of all contributions to it.