A note on the equivariant Dold-Thom theorem
type: Item
instance of: scholarly article
title: A note on the equivariant Dold-Thom theorem
zbMATH Open document ID: 1032.55010
DOI: 10.1016/S0022-4049(03)00029-X
author: Pedro F. dos Santos
published in: Journal of Pure and Applied Algebra
publication date: 2003-11-17T00:00:00Z
review text: A classical theorem by Dold and Thom states that for a based CW-complex \(X\) there is a natural isomorphism \(\pi_n({\mathbb Z}\otimes X)\cong \tilde{H}_n(X; {\mathbb Z})\) where \({\mathbb Z}\otimes X\) denotes the free abelian group on \(X\). This theorem was generalized by \textit{P. C. Lima-Filho} [Math. Z. 224, 567-601 (1997; Zbl 0882.55008)] for the Bredon homology \(\tilde{H}^G_n(X; {\mathbb Z})\) of a based \(G\)-CW-complex \(X\) with coefficients in the Mackey functor \(\underline{\mathbb Z}\) associated to \({\mathbb Z}\), where \(G\) is a finite group. The aim of the present paper is to generalize Lima-Filho's result for RO(\(G\))-graded equivariant homology functors \(\tilde{H}^G_V(X; \underline{M})\), where \(G\) denotes also a finite group and \(\underline{M}\) is the Mackey functor associated to a discrete \({\mathbb Z}[G]\)-module \(M\). For this purpose, Dold's and Thom's \(\mathbb{Z}\otimes X\) must be extended for \(M\) and \(X\) given above. The relevant topological group \(M\otimes X\) is defined as a quotient of the disjoint union of all \(M^n\times X^n\)'s, \(n \geq 0\). But as a \({\mathbb Z}[G]\)-module, this can simply be written as the direct sum of copies of \(M\) indexed over \(X-\{*\}\), where \(*\) denotes the base point of \(X\), and then the action of \(g \in G\) is given by \((g\cdot m)_x=g\cdot m_{g^{-1}\cdot x}\), where \(m_x\) denotes the \(x\)th coordinate of \(m \in M\otimes X\). The functor \(M\otimes -\) has useful homotopical properties for the arguments below in the categories of \(G\)-sets and \(G\)-spaces. Especially, it is essential that this transforms \(G\)-cofibre sequences into \(G\)-fibre sequences. Let \(V\) be a finite dimensional \(G\)-representation and \(S^V\) its one-point compactification. Then the set \(\{ M\otimes (S^V\wedge X) \}\), where \(V\) runs over an indexing set of a complete \(G\)-universe, defines a \(G\)-prespectrum with the structural maps \(S^W\wedge(M\otimes (S^V\wedge X)) \to M\otimes (S^{W…
reviewed by: Haruo Minami
Mathematics Subject Classification ID: 55P91
Mathematics Subject Classification ID: 55P42
Mathematics Subject Classification ID: 55N91
Mathematics Subject Classification ID: 55P20
Mathematics Subject Classification ID: 55P47
zbMATH DE Number: 2005430
zbMATH Keywords: equivariant Dold-Thom theorem
zbMATH Keywords: equivariant homotopy theory
zbMATH Keywords: RO(G)-graded homology theory
zbMATH Keywords: Lawson homology
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2005430