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A Shapiro lemma for diagrams of spaces with applications to equivariant topology. (English) Zbl 0853.55005

The authors study equivariant Bredon (co-)homology with twisted coefficients. Their formulation of the equivariant Whitehead theorem (Theorem 1.1) allows the fixed point sets as non(simply) connected. (For a more restricted version see [S. Illman, Equivariant singular homology and cohomology I, Mem. Am. Math. Soc. 156 (1975; Zbl 0297.55003)].)
The Shapiro lemma of the title addresses an isomorphism \[ H^*(\text{induced } (X), m)\cong H^* (X, \text{restricted } (M)) \] as well as a spectral sequence \[ H^p (X, R^q (\text{induced }(M)) \Rightarrow H^{p+q} (\text{restricted }(X), (M)). \] The setting is for general diagrams of spaces. Two useful appendices recollect categorical constructions used in the paper.

MSC:

55N25 Homology with local coefficients, equivariant cohomology
55N91 Equivariant homology and cohomology in algebraic topology

Citations:

Zbl 0297.55003
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References:

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