Geometry & Topology

Construction of 2–local finite groups of a type studied by Solomon and Benson

Ran Levi and Bob Oliver

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Abstract

A p–local finite group is an algebraic structure with a classifying space which has many of the properties of p–completed classifying spaces of finite groups. In this paper, we construct a family of 2–local finite groups, which are exotic in the following sense: they are based on certain fusion systems over the Sylow 2–subgroup of  Spin7(q) (q an odd prime power) shown by Solomon not to occur as the 2–fusion in any actual finite group. Thus, the resulting classifying spaces are not homotopy equivalent to the 2–completed classifying space of any finite group. As predicted by Benson, these classifying spaces are also very closely related to the Dwyer–Wilkerson space BDI(4).

Article information

Source
Geom. Topol., Volume 6, Number 2 (2002), 917-990.

Dates
Received: 22 October 2002
Accepted: 31 December 2002
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513882943

Digital Object Identifier
doi:10.2140/gt.2002.6.917

Mathematical Reviews number (MathSciNet)
MR1943386

Zentralblatt MATH identifier
1021.55010

Subjects
Primary: 55R35: Classifying spaces of groups and $H$-spaces
Secondary: 55R37: Maps between classifying spaces 20D06: Simple groups: alternating groups and groups of Lie type [See also 20Gxx] 20D20: Sylow subgroups, Sylow properties, $\pi$-groups, $\pi$-structure

Keywords
Classifying space $p$–completion finite groups fusion.

Citation

Levi, Ran; Oliver, Bob. Construction of 2–local finite groups of a type studied by Solomon and Benson. Geom. Topol. 6 (2002), no. 2, 917--990. doi:10.2140/gt.2002.6.917. https://projecteuclid.org/euclid.gt/1513882943


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References

  • M Aschbacher, A characterization of Chevalley groups over fields of odd order, Annals of Math. 106 (1977) 353–398
  • M Aschbacher, Finite group theory, Cambridge Univ. Press (1986)
  • D Benson, Cohomology of sporadic groups, finite loop spaces, and the Dickson invariants, Geometry and cohomology in group theory, London Math. Soc. Lecture notes ser. 252, Cambridge Univ. Press (1998) 10–23
  • P Bousfield, D Kan, Homotopy limits, completions and localizations, Lecture notes in math. 304, Springer–Verlag (1972)
  • C Broto, R Levi, B Oliver, Homotopy equivalences of $p$–completed classifying spaces of finite groups, Invent. math. (to appear)
  • C Broto, R Levi, B Oliver, The homotopy theory of fusion systems, preprint
  • C Broto, J Møller, Homotopy finite Chevalley versions of $p$–compact groups, (in preparation)
  • J Dieudonné, La géométrie des groupes classiques, Springer–Verlag (1963)
  • W Dwyer, C Wilkerson, A new finite loop space at the prime two, J. Amer. Math. Soc. 6 (1993) 37–64
  • W Dwyer, C Wilkerson, Homotopy fixed-point methods for Lie groups and finite loop spaces, Annals of Math. 139 (1994) 395–442
  • W Dwyer, C Wilkerson, The center of a $p$–compact group, The Čech centennial, Contemp. Math. 181 (1995) 119–157
  • E Friedlander, Etale homotopy of simplicial schemes, Princeton Univ. Press (1982)
  • E Friedlander, G Mislin, Cohomology of classifying spaces of complex Lie groups and related discrete groups, Comment. Math. Helv. 59 (1984) 347–361
  • D Goldschmidt, Strongly closed 2–subgroups of finite groups, Annals of Math. 102 (1975) 475–489
  • D Gorenstein, Finite groups, Harper & Row (1968)
  • S Jackowski, J McClure, B Oliver, Homotopy classification of self-maps of $BG$ via $G$–actions, Annals of Math. 135 (1992) 184–270
  • J Lannes, Sur les espaces fonctionnels dont la source est le classifiant d'un $p$–groupe abélien élémentaire, Publ. I.H.E.S. 75 (1992)
  • D Notbohm, On the 2–compact group DI(4), J. Reine Angew. Math. (to appear)
  • L Puig, Unpublished notes
  • L Smith, Homological algebra and the Eilenberg-Moore spectral sequence, Trans. Amer. Math. Soc. 129 (1967) 58–93
  • L Smith, Polynomial invariants of finite groups, A K Peters (1995)
  • R Solomon, Finite groups with Sylow 2–subgroups of type $.3$, J. Algebra 28 (1974) 182–198
  • M Suzuki, Group theory I, Springer–Verlag (1982)
  • D Taylor, The geometry of the classical groups, Heldermann Verlag (1992)
  • C Weibel, An introduction to homological algebra, Cambridge Univ. Press (1994)
  • C Wilkerson, A primer on the Dickson invariants, Proc. Northwestern homotopy theory conference 1982, Contemp. Math. 19 (1983) 421–434