Algebraic & Geometric Topology

A spectral sequence for fusion systems

Antonio Díaz Ramos

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Abstract

We build a spectral sequence converging to the cohomology of a fusion system with a strongly closed subgroup. This spectral sequence is related to the Lyndon–Hochschild–Serre spectral sequence and coincides with it for the case of an extension of groups. Nevertheless, the new spectral sequence applies to more general situations like finite simple groups with a strongly closed subgroup and exotic fusion systems with a strongly closed subgroup. We prove an analogue of a result of Stallings in the context of fusion preserving homomorphisms and deduce Tate’s p–nilpotency criterion as a corollary.

Article information

Source
Algebr. Geom. Topol., Volume 14, Number 1 (2014), 349-378.

Dates
Received: 13 December 2012
Revised: 22 May 2013
Accepted: 29 May 2013
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715804

Digital Object Identifier
doi:10.2140/agt.2014.14.349

Mathematical Reviews number (MathSciNet)
MR3158762

Zentralblatt MATH identifier
1297.55019

Subjects
Primary: 55T10: Serre spectral sequences
Secondary: 55R35: Classifying spaces of groups and $H$-spaces 20D20: Sylow subgroups, Sylow properties, $\pi$-groups, $\pi$-structure

Keywords
Lyndon–Hochschild–Serre spectral sequence fusion system strongly closed subgroup Tate's nilpotency criterion

Citation

Díaz Ramos, Antonio. A spectral sequence for fusion systems. Algebr. Geom. Topol. 14 (2014), no. 1, 349--378. doi:10.2140/agt.2014.14.349. https://projecteuclid.org/euclid.agt/1513715804


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References

  • M Aschbacher, Normal subsystems of fusion systems, Proc. Lond. Math. Soc. 97 (2008) 239–271
  • C Broto, N Castellana, J Grodal, R Levi, B Oliver, Extensions of $p$–local finite groups, Trans. Amer. Math. Soc. 359 (2007) 3791–3858
  • C Broto, R Levi, B Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16 (2003) 779–856
  • K S Brown, Cohomology of groups, Grad. Texts in Math. 87, Springer, New York (1982)
  • J Cantarero, J Scherer, A Viruel, Nilpotent $p$–local finite groups (2011)
  • H Cartan, S Eilenberg, Homological algebra, Princeton University Press (1956)
  • D A Craven, Control of fusion and solubility in fusion systems, J. Algebra 323 (2010) 2429–2448
  • A Díaz, A Glesser, S Park, R Stancu, Tate's and Yoshida's theorems on control of transfer for fusion systems, J. Lond. Math. Soc. 84 (2011) 475–494
  • A Díaz, A Ruiz, A Viruel, All $p$–local finite groups of rank two for odd prime $p$, Trans. Amer. Math. Soc. 359 (2007) 1725–1764
  • L Evens, The cohomology of groups, Clarendon Press, Oxford (1991)
  • R J Flores, R M Foote, Strongly closed subgroups of finite groups, Adv. Math. 222 (2009) 453–484
  • J H Long, The cohomology rings of the special affine group of $\mathbb{F}_p^2$ and of $\mathrm{PSL}(3,p)$, PhD thesis, University of Maryland (2008) Available at \setbox0\makeatletter\@url http://search.proquest.com//docview/304560975 {\unhbox0
  • S Mac Lane, Homology, Springer, Berlin (1995)
  • S Park, Realizing a fusion system by a single finite group, Arch. Math. $($Basel$)$ 94 (2010) 405–410
  • K Ragnarsson, Classifying spectra of saturated fusion systems, Algebr. Geom. Topol. 6 (2006) 195–252
  • R Solomon, R Stancu, Conjectures on finite and $p$–local groups, L'Enseignement Mathématique 54 (2008) 171–176
  • J Stallings, Homology and central series of groups, J. Algebra 2 (1965) 170–181
  • J Tate, Nilpotent quotient groups, Topology 3 (1964) 109–111
  • P Webb, A guide to Mackey functors, from: “Handbook of algebra”, (M Hazewinkel, editor), North-Holland, Amersterdam (2000) 805–836