Topological groups and Mackey functors
Let \(M\) be a Mackey functor for a finite group \(G\) with values in the category of abelian groups. Given a pointed \(G\)-set \(C\) with base point \(x_0\) one lets \(F(C, M)\) denote the set consisting of maps \(x \mapsto u(x) \in M(G/G_x)\) for every \(x\) in \(C\) such that \(u(x_o)=0\) and \(u(x)=0\) for almost all \(x \in C\). This becomes a topological abelian \(G\)-group in a natural way. The action of \(G\) is given by \((g\cdot u)(x)=M_*(R_{g^{-1}}(u(g^{-1}x))\). Denote by \(F(C, M)^G\) the subgroup of fixed points of \(F(C, M)\) under this action. The authors show that two kinds of functorial structures on it can be found. Denote the resulting functors by \(\bar{F}^G(-, M)\) and \(F^G(-, M)\), then one finds that they are covariant and hence can be extended to functors defined on simplicial pointed \(G\)-sets. This allows one to define topological spaces \(\bar{F}^G(X, M)\) and \(F^G(X, M)\) for a pointed \(G\)-space \(X\) as the geometric realizations of simplicial groups \(\bar{F}^G(\mathcal{S}(X), M)\) and \(F^G(\mathcal{S}(X), M)\) where \(\mathcal{S}(X)\) denotes the singular simplicial pointed \(G\)-set of \(X\), both of which become naturally topological abelian groups. The second one was already studied in the authors' previous paper [Topology Appl. 154, No.~15, 2826--2848 (2007); erratum ibid. 156, No. 8, 1609-1613 (2009; Zbl 1130.55004)]. Instead, here one has one more new topological group \(\mathbb{F}^G(X, M)\), which is defined as \(F(X^\delta, M)^G\) with a certain identification topology where \(X^\delta\) denotes the underlying pointed \(G\)-space of \(X\). It is proved that this group becomes homotopy equivalent to \(F^G(X, M)\) under the conditions that \(M\) is homological and \(X\) has the homotopy type of a \(G\)-CW-complex. The main result of this paper states that the homotopy groups of \(\bar{F}^G(X, M)\) are isomorphic to the Bredon-Illman equivariant homology of \(X\) with coefficients in a coefficient system \(\bar{M}_*\) associated to \(M\) (Theorem (7.5)) and if the conditions stated above are satisfied then the homotopy groups of \(\mathbb{F}^G(X, M)\) are isomorphic to the Bredon-Illman equivariant homology of \(X\) with coefficients in the covariant part \(M_*\) of \(M\) (Theorem (7.6)). Moreover, the authors study the the transfer \(t_P^G : \mathbb{F}^G(X, M) \to \mathbb{F}^G(E, M)\) for any finite covering \(G\)-map \(p : E \to X\), and prove that in the case of a \((G, \Gamma)\)-locally trivial bundle the composite \(p_*^G\circ t_*^G\) is given by multiplication by \(|\Gamma|\) (Theorem (8.11)).
scientific article; zbMATH DE number 5794607