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Finiteness conditions in Bredon cohomology. (English) Zbl 0748.55003
Let \(G\) be a finite group. If \(q\) is a natural number and \(A\) is a \(\mathbb{Z}[G]\) module such that for every prime \(p\) and every \(p\)-subgroup \(H\) of \(G\), \(H^{q+1}(H,A)=H^ q(H,A)=0\), then \(H^ n(G,A)=0\) for every \(n\geq q\). This is a well known result in cohomology theory of groups which is generalized in the paper under discussion.
Theorem. Let \(K\) be a \(G\)-CW complex and let \(M\) be a \(G\)-coefficient system as defined by G. E. Bredon [Equivariant cohomology theories (Lect. Notes Math. 34) (1967; Zbl 0162.272)]. Let \(n\) be a natural number. Suppose that, for every \(m\geq n\) and for every subgroup \(H\) of \(G\), the equivariant Bredon cohomology groups \(H^ m_ G(K^ H,K^{>H};M)\) vanish. Let \(G'\) be a subgroup of \(G\) such that, for every prime \(p\) and every \(p\)-subgroup \(H\) of \(G'\), the cohomology groups \(H^ m_ G(K\times_{K/G}K/H,M)\) vanish whenever \(n\leq m\leq n+1=4l_ p(G')\). Then, for all \(m\geq n\), \(H^ m_ G(K\times_{K/G}K/G',M)=0\).
In this theorem \(l_ p(G)=\log_ p| G_ p|\) where \(G_ p\) is the \(p\) Sylow subgroup of \(G\). Furthermore, \(K\times_{K/G}K/H\) is obtained as the fibre product of the two projection maps \(K\to K/G\) and \(K/H\to K/G\).

MSC:
55N91 Equivariant homology and cohomology in algebraic topology
20J06 Cohomology of groups
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References:
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[6] J. Słomińska, Dimension in Bredon cohomology, In preparation.
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