Topological groups and Mackey functors
type: Item
instance of: scholarly article
title: Topological groups and Mackey functors
zbMATH Open document ID: 1206.55009
author: Carlos Prieto
author: Marcelo Aguilar
published in: Boletín de la Sociedad Matemática Mexicana. Third Series
publication date: 2010-10-01T00:00:00Z
review text: 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 \(\bar{M}_*\) assoc…
reviewed by: Haruo Minami
Mathematics Subject Classification ID: 55N91
Mathematics Subject Classification ID: 55R91
Mathematics Subject Classification ID: 14F43
Mathematics Subject Classification ID: 57M10
zbMATH DE Number: 5794607
zbMATH Keywords: Equivariant homology
zbMATH Keywords: homotopy groups
zbMATH Keywords: Mackey functors
zbMATH Keywords: equivariant covering maps
zbMATH Keywords: transfer
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5794607