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Euler classes and Bredon cohomology for groups with restricted families of finite subgroups. (English) Zbl 1294.20066

Let \(\Gamma\) be a group. The author explores different invariants associated to \(\Gamma\) that are defined in terms of the family of its finite subgroups. Let \(\underline E\Gamma\) denote the classifying space for proper actions for \(\Gamma\), and \(\mathcal F\) the poset of finite subgroups of \(\Gamma\). Moreover, let \(\underline{\mathrm{cd}}_R\Gamma\) denote the smallest length of a projective resolution of the contravariant trivial Bredon module \(R\) for the family of finite subgroups of \(\Gamma\).
The first result in this work is the following: Theorem A. Let \(\Gamma\) be a group having all of its finite subgroups nilpotent and of bounded order. Then \[ \underline{\mathrm{cd}}_R\Gamma\leq\max_{H\in\mathcal F}[\mathrm{pd}_{R\Gamma}(B(RW(H)))+r(W(H))], \] where \(W(H)=N_\Gamma(H)/H\), \(r(W(H))\) is the largest rank of a finite elementary Abelian subgroup of \(W(H)\) and \(B(R\Gamma)=\{f\colon\Gamma\to R\mid f(\Gamma)\text{ is finite}\}\).
The equivariant Euler class, \(\chi(\underline E\Gamma)\) is defined under the pertinent finiteness conditions. The author also gives a computation of the coefficient of the term \([\Gamma/1]\) in the equivariant Euler class in terms of certain elementary Abelian subgroups of \(\Gamma\), their normalizers and the equivariant Euler class of their centralizers.
Finally, the author gives some relations of Poincaré duality and Bredon Poincaré duality, more precisely, Theorem C. Let \(\Gamma=K\ltimes G\) where \(G\) is torsion free and \(K\) a \(p\)-group and let \(F\) be a field of prime characteristic \(p\). Then
\(G\) is Poincaré duality over \(F\) if and only if \(\Gamma\) is Bredon Poincaré duality over \(F\). In this case \(\underline{\mathrm{cd}}_F\Gamma=\mathrm{cd}_FG\).
For \(p=2\), there are examples where \(G\) is duality over \(F\) but \(\Gamma\) is not Bredon duality over \(F\).

MSC:

20J05 Homological methods in group theory
20J06 Cohomology of groups
18G20 Homological dimension (category-theoretic aspects)
18G35 Chain complexes (category-theoretic aspects), dg categories
55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology
57P10 Poincaré duality spaces
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References:

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