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Nullification functors and the homotopy type of the classifying space for proper bundles. (English) Zbl 1092.55012
Given two spaces \(A\) and \(X\), \(X\) is \(A\)-null if the mapping space \(\text{map}(A,X)\) is homotopy equivalent to \(X\) via the inclusion of the constant maps, \(X\hookrightarrow \text{map}(A,X)\). The \(A\)-nullification of a space is a functor on spaces that assigns to \(X\) an \(A\)-null space \(P_A(X)\) together with a universal map \(X\to P_A(X)\) which induces a weak homotopy equivalence \(\text{map}(P_A(X),Y) \simeq \text{map}(X,Y)\) for every \(A\)-null space \(Y\). Thus \(P_A\) ‘makes trivial the structure of \(X\) that depends on \(A\).’
Let \(G\) denote a discrete group throughout. The paper relates nullification with respect to \(W_\infty = \bigvee_p B\mathbb{Z}_p\), where the index is over the prime numbers, and the classifying space for proper actions of a group \(G\), denoted \(\underline{\text{B}}G\). The space \(\underline{\text{B}}G\) enjoys the universal property that if \(G\) acts on \(X\) with isotropy groups all finite, then there is a \(G\)-map \(X\to \underline{\text{E}}G\) unique up to \(G\) homotopy. The space \(\underline{\text{B}}G = \underline{\text{E}}G/G\). The main theorem shows that \(P_{W_\infty}(BG) \simeq \underline{\text{B}}G\).
Based on this equivalence, the author explores the properties of the classifying space for proper actions. As a functor \(\underline{\text{B}}\) takes products to products, and certain pushouts of groups to the corresponding pushout of classifying spaces. Using the properties of \(\bigvee_p B\mathbb{Z}_p\) and nullification, there are results for the homotopy type of \(\underline{\text{B}}G\) for \(G\) locally finite, supersoluble, or one of the wallpaper groups. Since \(\underline{\text{B}}G\) and \(BG\) are realized as realizations of nerves of small categories, the analysis of \(BG \to \underline{\text{B}}G\) can be studied by simplicial means, and the author determines the homotopy fibre of this map to end the paper.
MSC:
55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology
55P60 Localization and completion in homotopy theory
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