### Abstract

We show that if an inclusion of finite groups Ha parts per thousand currency signG of index prime to p induces a homeomorphism of mod p cohomology varieties, or equivalently an F-isomorphism in mod p cohomology, then H controls p-fusion in G, if p is odd. This generalizes classical results of Quillen who proved this when H is a Sylow p-subgroup, and furthermore implies a hitherto difficult result of Mislin about cohomology isomorphisms. For p=2 we give analogous results, at the cost of replacing mod p cohomology with higher chromatic cohomology theories.

The results are consequences of a general algebraic theorem we prove, that says that isomorphisms between p-fusion systems over the same finite p-group are detected on elementary abelian p-groups if p odd and abelian 2-groups of exponent at most 4 if p=2.

Original language | English |
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Pages (from-to) | 491-507 |

Number of pages | 16 |

Journal | Inventiones Mathematicae |

Volume | 197 |

Issue number | 3 |

Early online date | 13 Nov 2013 |

DOIs | |

Publication status | Published - Sep 2014 |

### Keywords

- classifying-spaces
- isomorphisms
- homology
- ring
- nilpotency
- spectrum
- maps

### Cite this

*Inventiones Mathematicae*,

*197*(3), 491-507. https://doi.org/10.1007/s00222-013-0489-5