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Equivalences of Classifying Spaces Completed at the Prime Two
Bob Oliver, Institut Galilée, Villetaneuse, France

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$65
Individual Members: US$39
Institutional Members: US$52
Order Code: MEMO/180/848
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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.

Table of Contents

  • 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
  • Sproadic groups
  • Computations of \(\mathrm{lim}^1(\mathcal{Z}_G)\)
  • Bibliography
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