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Representing Bredon cohomology with local coefficients. (English) Zbl 1318.55005
This paper continues the study of representability for Bredon cohomology with local coefficients. The two main theorems describe Bredon cohomology for a discrete group \(G\) first in terms of crossed complexes and second in terms of equivariant parametrized spectra. Note that the equivariant spectra used are naïve \(G\)-spectra, that is, they are indexed over a trivial universe.
A crossed complex can be (roughly) described as a chain complex of modules over a groupoid. Crossed complexes encode the algebraic structure of the sequence \(\Pi(X)=\pi_n(X_n, X_{n-1},x_0)\), where \(\{X_n \}_{n \geqslant 0}\) is the skeletal filtration of a space \(X\). Furthermore, crossed complexes can be used to give a representability result for cohomology with local coefficients. The first main result extends this to the equivariant setting. Given an equivariant local coefficent system \(M\) (with extra structure) the authors construct a series of equivariant crossed complexes \(\chi_G(M,n)\) such that the set of derived maps from the equivariant analogue of \(\Pi(X)\) to \(\chi_G(M,n)\) is the \(n^{th}\)-Bredon cohomology group with coefficients given by \(M\).
Parameterised spectra are spectra with a reference map to a base space \(B\). They can be used to give a representing result for Bredon cohomology with local coefficients. That is, given a local coefficient system, there is a parameterised spectrum \(E\) such that \(E\)-cohomology is Bredon cohomology with coefficients in that system. The second main result of this paper generalises this to the equivariant setting, using classifying spaces of equivariant crossed complexes to give an explicit construction of the representing equivariant parameterised spectrum.

55N25 Homology with local coefficients, equivariant cohomology
55N91 Equivariant homology and cohomology in algebraic topology
55P42 Stable homotopy theory, spectra
55P91 Equivariant homotopy theory in algebraic topology
55Q91 Equivariant homotopy groups
55T99 Spectral sequences in algebraic topology
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