Algebraic & Geometric Topology

The normaliser decomposition for $p$–local finite groups

Assaf Libman

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Abstract

We construct an analogue of the normaliser decomposition for p–local finite groups (S,,) with respect to collections of –centric subgroups and collections of elementary abelian subgroups of S. This enables us to describe the classifying space of a p–local finite group, before p–completion, as the homotopy colimit of a diagram of classifying spaces of finite groups whose shape is a poset and all maps are induced by group monomorphisms.

Article information

Source
Algebr. Geom. Topol., Volume 6, Number 3 (2006), 1267-1288.

Dates
Received: 4 February 2005
Accepted: 29 June 2006
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796577

Digital Object Identifier
doi:10.2140/agt.2006.6.1267

Mathematical Reviews number (MathSciNet)
MR2253446

Zentralblatt MATH identifier
1128.55006

Subjects
Primary: 55R35: Classifying spaces of groups and $H$-spaces 55P05: Homotopy extension properties, cofibrations

Keywords
homology decomposition $p$–local finite groups

Citation

Libman, Assaf. The normaliser decomposition for $p$–local finite groups. Algebr. Geom. Topol. 6 (2006), no. 3, 1267--1288. doi:10.2140/agt.2006.6.1267. https://projecteuclid.org/euclid.agt/1513796577


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References

  • A K Bousfield, D M Kan, Homotopy limits, completions and localizations, Springer, Berlin (1972)
  • C Broto, N Castellana, J Grodal, R Levi, B Oliver, Subgroup families controlling $p$–local groups, J. London Math. Soc. to appear
  • C Broto, R Levi, B Oliver, Homotopy equivalences of $p$–completed classifying spaces of finite groups, Invent. Math. 151 (2003) 611–664
  • C Broto, R Levi, B Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16 (2003) 779–856
  • W G Dwyer, Homology decompositions for classifying spaces of finite groups, Topology 36 (1997) 783–804
  • W G Dwyer, D M Kan, A classification theorem for diagrams of simplicial sets, Topology 23 (1984) 139–155
  • P G Goerss, J F Jardine, Simplicial homotopy theory, Progress in Mathematics 174, Birkhäuser Verlag, Basel (1999)
  • J Hollender, R M Vogt, Modules of topological spaces, applications to homotopy limits and $E_{\infty}$ structures, Arch. Math. $($Basel$)$ 59 (1992) 115–129
  • A Libman, A Minami–Webb splitting of classifying spaces of finite groups and the exotic examples of Ruiz and Viruel, preprint
  • A Libman, A Viruel, On the homotopy type of the non-completed classifying space of a $p$–local finite group, preprint
  • S Mac Lane, Categories for the working mathematician, Graduate Texts in Mathematics 5, Springer, New York (1998)
  • J P May, Simplicial objects in algebraic topology, Van Nostrand Mathematical Studies, No. 11, D. Van Nostrand Co., Princeton, N.J.-Toronto, Ont.-London (1967)
  • L Puig, Unpublished notes
  • D Quillen, Higher algebraic $K$–theory. I, from: “Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972)”, Springer, Berlin (1973) 85–147. Lecture Notes in Math., Vol. 341
  • A Ruiz, A Viruel, The classification of $p$–local finite groups over the extraspecial group of order $p^3$ and exponent $p$, Math. Z. 248 (2004) 45–65
  • J Słomińska, Homotopy colimits on E-I-categories, from: “Algebraic topology Poznań 1989”, Lecture Notes in Math. 1474, Springer, Berlin (1991) 273–294
  • R W Thomason, Homotopy colimits in the category of small categories, Math. Proc. Cambridge Philos. Soc. 85 (1979) 91–109