Equivalences of Classifying Spaces Completed at the Prime Two
- ISBN-10
- 0821838288
- ISBN-13
- 9780821838280
- Category
- Classifying spaces
- Pages
- 102
- Language
- English
- Published
- 2006
- Publisher
- American Mathematical Soc.
- Authors
- Bob Oliver, Robert Oliver
Description
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.
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