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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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