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Equivalences of Classifying Spaces Completed at the Prime Two
Bob Oliver, Institut Galilée, Villetaneuse, France
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Memoirs of the American Mathematical Society
2006; 102 pp; softcover
Volume: 180
ISBN-10: 0-8218-3828-8
ISBN-13: 978-0-8218-3828-0
List Price: US$61 Individual Members: US$36.60
Institutional Members: US\$48.80
Order Code: MEMO/180/848

We prove here the Martino-Priddy conjecture at the prime $$2$$: the $$2$$-completions of the classifying spaces of two finite groups $$G$$ and $$G'$$ are homotopy equivalent if and only if there is an isomorphism between their Sylow $$2$$-subgroups which preserves fusion. This is a consequence of a technical algebraic result, which says that for a finite group $$G$$, the second higher derived functor of the inverse limit vanishes for a certain functor $$\mathcal{Z}_G$$ on the $$2$$-subgroup orbit category of $$G$$. The proof of this result uses the classification theorem for finite simple groups.

• Introduction
• Higher limits over orbit categories
• Reduction to simple groups
• A relative version of $$\Lambda$$-functors
• Subgroups which contribute to higher limits
• Alternating groups
• Groups of Lie type in characteristic two
• Classical groups of Lie type in odd characteristic
• Exceptional groups of Lie type in odd characteristic
• Computations of $$\mathrm{lim}^1(\mathcal{Z}_G)$$