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Equivalences of Classifying Spaces Completed at the Prime Two
 
Bob Oliver Institut Galilée, Villetaneuse, France
Front Cover for Equivalences of Classifying Spaces Completed at the Prime Two
Available Formats:
Electronic ISBN: 978-1-4704-0452-9
Product Code: MEMO/180/848.E
List Price: $65.00
MAA Member Price: $58.50
AMS Member Price: $39.00
Front Cover for Equivalences of Classifying Spaces Completed at the Prime Two
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  • Front Cover for Equivalences of Classifying Spaces Completed at the Prime Two
  • Back Cover for Equivalences of Classifying Spaces Completed at the Prime Two
Equivalences of Classifying Spaces Completed at the Prime Two
Bob Oliver Institut Galilée, Villetaneuse, France
Available Formats:
Electronic ISBN:  978-1-4704-0452-9
Product Code:  MEMO/180/848.E
List Price: $65.00
MAA Member Price: $58.50
AMS Member Price: $39.00
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1802006; 102 pp
    MSC: Primary 55; Secondary 20;

    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
     
     
    • Chapters
    • Introduction
    • 1. Higher limits over orbit categories
    • 2. Reduction to simple groups
    • 3. A relative version of $\Lambda $-functors
    • 4. Subgroups which contribute to higher limits
    • 5. Alternating groups
    • 6. Groups of Lie type in characteristic two
    • 7. Classical groups of Lie type in odd characteristic
    • 8. Exceptional groups of Lie type in odd characteristic
    • 9. Sporadic groups
    • 10. Computations of $\lim ^1(\mathcal {Z}_G)$
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Volume: 1802006; 102 pp
MSC: Primary 55; Secondary 20;

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.

  • Chapters
  • Introduction
  • 1. Higher limits over orbit categories
  • 2. Reduction to simple groups
  • 3. A relative version of $\Lambda $-functors
  • 4. Subgroups which contribute to higher limits
  • 5. Alternating groups
  • 6. Groups of Lie type in characteristic two
  • 7. Classical groups of Lie type in odd characteristic
  • 8. Exceptional groups of Lie type in odd characteristic
  • 9. Sporadic groups
  • 10. Computations of $\lim ^1(\mathcal {Z}_G)$
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