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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 Click above image for expanded view 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.

• 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
• 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
• 10. Computations of $\lim ^1(\mathcal {Z}_G)$