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Smith theory and quasi-periodicity in Bredon cohomology. (English) Zbl 0762.55005
Let \(G\) be a finite group. A \(CW\)-complex \(K\) with a given action of \(G\) is said to be a \(G\)-\(CW\)-complex, if for every subgroup \(H\) of \(G\), the fixed point set \(K^ H\) is a subcomplex of \(K\). In this paper, the author studies the equivariant cohomology theory of \(G\)-\(CW\)-complexes by using Bredon cohomology. Thus the author was able to generalize some well-known results in the classical Smith theory. For intance, the following special case of his results will yield Theorem 5.4 in Chapter III of G. E. Bredon’s book: Introduction to compact transformation groups (1972; Zbl 0246.57017)]. Let \(K\) be a \(G\)-\(CW\)-complex. Assume that \(A\) is an abelian group, and that \(q\) and \(m\) are natural numbers such that \(q>m\). Suppose that, for every subgroup \(H\) of \(G\), \(H^ n(K/H,A)=0\), whenever \(n=q\), \(q-1\), and that \(H^ n(K,A)=0\) whenever \(m\leq n\leq q\). Then \(H^ n(K/G,A)=0\) whenever \(m\leq n\leq q\).
Reviewer: H.T.Ku (Amherst)

MSC:
55N91 Equivariant homology and cohomology in algebraic topology
55M35 Finite groups of transformations in algebraic topology (including Smith theory)
57S17 Finite transformation groups
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References:
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