A note on the equivariant Dold-Thom theorem
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 \(S^W\wedge(M\otimes (S^V\wedge X)) \to M\otimes (S^{W+V}\wedge X)\) given by making use of the \({\mathbb Z}[G]\)-module structure of \(M\). Let \(M\otimes \Sigma^\infty X\) denote the associated \(G\)-spectrum. The author first shows that particularly \(M\otimes \Sigma^\infty S^0\) becomes an Eilenberg-Mac Lane spectrum \(H\underline{M}\), so that \(\tilde{H}^G_V(X; \underline{M}) \cong \pi^G_V((M\otimes \Sigma^\infty S^0)\wedge X)\) follows. Here \ \(\pi^G_V(X)\) denotes the set of equivariant homotopy classes \([S^V, X]_G\). Next the author considers a natural map \((M\otimes S^V)\wedge X \to M\otimes (S^V\wedge X)\). By assembling these maps one obtains a map of \(G\)-spectra \((M\otimes \Sigma^\infty S^0)\wedge X \to M\otimes \Sigma^\infty X\). The author proves that the zero component of this map is a \(G\)-homotopy equivalence. It is also proved that the inclusion of \(M\otimes X\) into the zero component of \(M\otimes \Sigma^\infty X\) is a \(G\)-homotopy equivalence. Thus one finds that putting these two \(G\)-homotopy equivalences together yields the desired equivalence \(\pi^G_V(M\otimes X)\cong \tilde{H}^G_V(X; \underline{M})\) (Theorem 1.1). An application of this result to study of algebraic cycles on real projective varieties is given in the final section.
scientific article; zbMATH DE number 2005430
scientific article