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Unstable homotopy classification of \(BG_ p^ \wedge\). (English) Zbl 0855.55008

The space \(BG^\wedge_p\) is the \(p\)-completion of \(BG\), the classifying space of a finite group \(G\). It is first shown how to construct this space as a homotopy colimit using the Frobenius category \({\mathcal P} (g)\) which has as objects the \(p\)-subgroups of \(G\) and as morphisms the homomorphisms induced by conjugation by elements of \(G\). Then a classification result is proved which determines \(BG^\wedge_p\) upto homotopy equivalence from its \(p\)-local data, i.e., conjugacy classes of \(p\)-subgroups of \(G\) and their Weyl groups. As a corollary necessary and sufficient conditions are obtained for two \(BG^\wedge_p\) spaces to be homotopically equivalent. There exist maps between two \(BG^\wedge_p\) spaces which are not induced by homomorphisms. This is true even for equivalences; consequently the question arises whether a homotopy equivalence between two \(BG^\wedge_p\) spaces can be realised on the group level as an alternating sequence of homomorphisms each inducing an equivalence. Examples are given to show that the answer is negative.

MSC:

55P60 Localization and completion in homotopy theory
55R40 Homology of classifying spaces and characteristic classes in algebraic topology
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References:

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