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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:
 [1] Bredon, G.E., Equivariant cohomology theory, () · Zbl 0162.27202 [2] Bröcker, T., Singulare definition der äequivarianten Bredon homologie, Manuscripta math., 5, 91-102, (1971) · Zbl 0213.49902 [3] Brown, K.S., Cohomology of groups, (1982), Springer Berlin · Zbl 0367.18012 [4] tom Dieck, T., Orbittypen und äequivariante homologie, Arch. math., 23, 307-317, (1972) · Zbl 0252.55003 [5] Gorenstein, D., Finite groups, (1968), Harper & Row New York · Zbl 0185.05701 [6] J. Słomińska, Dimension in Bredon cohomology, In preparation.
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