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Bredon cohomology

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Idea

What is called Bredon cohomology after (Bredon 67) is the flavor of 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, insteadof 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 /BGL fpweGTopPSh (Orb G). \mathbf{H}_{/\mathbf{B}G} \coloneqq L_{fpwe} G Top \simeq PSh_\infty(Orb_G) \,.

A spectrum object EStab(H /BG)E \in Stab(\mathbf{H}_{/\mathbf{B}G}) in the (∞,1)-topos H /BG\mathbf{H}_{/\mathbf{B}G} 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 /BG\mathbf{H}_{/\mathbf{B}G}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 BG K(n,A) \in \mathbf{H}_{\mathbf{B}G}

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 /BG\mathbf{H}_{/\mathbf{B}G} with coefficients in K(n,A)K(n,A):

H G n(X,A)π 0H /BG(X,A) H_G^n(X,A) \simeq \pi_0 \mathbf{H}_{/\mathbf{B}G}(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 appears for instance in (Greenlees-May, p. 10).

But what (Bredon 67) really wrote down 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.

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

Borel equivariant cohomology\leftarrowgeneral (Bredon) equivariant cohomology\rightarrownon-equivariant cohomology with homotopy fixed point coefficients
H(X G,A)\mathbf{H}(X_G, A)trivial action on coefficients AA[X,A] G[X,A]^Gtrivial action on domain space XXH(X,A G)\mathbf{H}(X, A^G)

References

The original text is

  • Glen Bredon, Equivariant cohomology theories, Springer Lecture Notes in Mathematics Vol. 34. 1967.

Reviews include

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

  • Paolo Masulli, section 2 of Equivariant homotopy: KRKR-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 CatCat-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 KK-theory which is closely related to the Bredon cohomology (rather than usual equivariant cohomology):

Revised on April 16, 2014 06:03:26 by Urs Schreiber (145.116.129.4)