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Group cohomology and control of p-fusion

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Abstract

We show that if an inclusion of finite groups H≤G 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.

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Acknowledgements

Our interest in Theorem A was piqued by a discussion during the problem session at the August 2011 workshop on Homotopical Approaches to Group Actions in Copenhagen with Peter Symonds and others, and also stimulated by [11]. We thank Mike Hopkins for pointers concerning the relationship between Theorem C and classical stable homotopy theory, explained in Remark 4.5, and Lucho Avramov, Nick Kuhn, and Neil Strickland for other literature references.

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Authors and Affiliations

  1. Institute of Mathematics, University of Aberdeen, Aberdeen, AB24 3UE, UK

    David J. Benson

  2. Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark

    Jesper Grodal & Ellen Henke

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  1. David J. Benson
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  2. Jesper Grodal
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Correspondence to Jesper Grodal.

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Benson, D.J., Grodal, J. & Henke, E. Group cohomology and control of p-fusion. Invent. math. 197, 491–507 (2014). https://doi.org/10.1007/s00222-013-0489-5

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