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Equivariant weak \(n\)-equivalences. (English) Zbl 0932.55010
This paper starts by giving a good account of the nonequivariant case together with a list of equivalent conditions for weak \(n\)-equivalences constructed by using homology with local coefficients. The main theorem in this paper provides a list of equivalent conditions for a map \(f:X\rightarrow Y\) between path connected CW-spaces to be a \(G\)-equivariant \(n\) equivalence where \(G\) is a discrete group and the fixed point spaces \(X^H\) and \(Y^H\) are path connected CW-spaces and satisfy that the groups \(\pi_1(X^H)\) and \(\pi_1(Y^H)\) are complete and the groups \(\Gamma_\pi\pi_k(Y^H,X^H)\) and \(\Gamma_\pi\pi_k(X^H)\) are trivial for \(k=2,\ldots,n\) and for every group \(H\subseteq G\).
MSC:
55P91 Equivariant homotopy theory in algebraic topology
55P10 Homotopy equivalences in algebraic topology
55N91 Equivariant homology and cohomology in algebraic topology
55U35 Abstract and axiomatic homotopy theory in algebraic topology
55Q91 Equivariant homotopy groups
57S15 Compact Lie groups of differentiable transformations
55P15 Classification of homotopy type
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