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The homotopy theory of fusion systems


Authors: Carles Broto, Ran Levi and Bob Oliver
Journal: J. Amer. Math. Soc. 16 (2003), 779-856
MSC (2000): Primary 55R35; Secondary 55R40, 20D20
DOI: https://doi.org/10.1090/S0894-0347-03-00434-X
Published electronically: July 21, 2003
MathSciNet review: 1992826
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Abstract: We define and characterize a class of $p$-complete spaces $X$ which have many of the same properties as the $p$-completions of classifying spaces of finite groups. For example, each such $X$ has a Sylow subgroup $BS\longrightarrow X$, maps $BQ\longrightarrow X$ for a $p$-group $Q$ are described via homomorphisms $Q\longrightarrow S$, and $H^*(X;\mathbb{F} _p)$ is isomorphic to a certain ring of ``stable elements'' in $H^*(BS;\mathbb{F} _p)$. These spaces arise as the ``classifying spaces'' of certain algebraic objects which we call ``$p$-local finite groups''. Such an object consists of a system of fusion data in $S$, as formalized by L. Puig, extended by some extra information carried in a category which allows rigidification of the fusion data.


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  • [Alp] J. Alperin, Local representation theory, Cambridge Univ. Press (1986). MR 87i:20002
  • [AB] J. Alperin & M. Broué, Local methods in block theory, Annals of Math. 110 (1979), 143-157.MR 80f:20010
  • [AT] E. Artin & J. Tate, A note on finite ring extensions, J. Math. Soc. Japan 3 (1951), 74-77.MR 13:427c
  • [AM] M. Atiyah & I. Macdonald, Introduction to commutative algebra, Adison-Wesley (1969).MR 39:4129
  • [Be] 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.MR 2001i:55017
  • [BK] P. Bousfield & D. Kan, Homotopy limits, completions, and localizations, Lecture notes in math. 304, Springer-Verlag (1972). MR 51:1825
  • [Bd] G. Bredon, Sheaf theory, McGraw Hill (1967).MR 36:4552
  • [BrK] C. Broto & N. Kitchloo, Classifying spaces of Kac-Moody groups, Math. Z. 240 (2002), 621-649.
  • [BL] C. Broto & R. Levi, On spaces of self homotopy equivalences of $p$-completed classifying spaces of finite groups and homotopy group extensions, Topology 41 (2002), 229-255.MR 2002j:55013
  • [BLO] C. Broto, R. Levi, & B. Oliver, Homotopy equivalences of $p$-completed classifying spaces of finite groups, Invent. Math. 151 (2003), 611-664.
  • [BrM] C. Broto & J. Møller, Homotopy finite Chevalley versions of $p$-compact groups, preprint, 2003.
  • [Bw] K. Brown, Cohomology of groups, Springer-Verlag (1982). MR 83k:20002
  • [Atl] J. Conway, R. Curtis, S. Norton, R. Parker, & R. Wilson, Atlas of finite groups, Oxford Univ. Press (1985).MR 88g:20025
  • [Dw] W. Dwyer, The centralizer decomposition of $BG$, in: Algebraic topology: new trends in localization and periodicity, Progr. Math. 136, Birkhäuser (1996), 167-184.MR 97i:55028
  • [DK] W. Dwyer & D. Kan, Realizing diagrams in the homotopy category by means of diagrams of simplicial sets, Proc. Amer. Math. Soc. 91 (1984), 456-460.MR 86c:55010b
  • [DK2] W. Dwyer & D. Kan, Centric maps and realizations of diagrams in the homotopy category, Proc. Amer. Math. Soc. 114 (1992), 575-584.MR 92e:55011
  • [DW1] W. Dwyer & C. Wilkerson, A cohomology decomposition theorem, Topology 31 (1992), 433-443.MR 93h:55008
  • [EP] L. Evens & S. Priddy, The ring of universally stable elements, Quart. J. Math. Oxford 40 (1989), 399-407.MR 90k:20089
  • [GZ] P. Gabriel & M. Zisman, Calculus of fractions and homotopy theory, Springer-Verlag (1967).MR 35:1019
  • [GJ] P. Goerss & J. Jardine, Simplicial homotopy theory, Birkhäuser (1999).MR 2001d:55012
  • [Go] D. Gorenstein, Finite groups, Harper & Row (1968). MR 38:229
  • [GL] D. Gorenstein & R. Lyons, The local structure of finite groups of characteristic 2 type, Memoirs Amer. Math. Soc. 276 (1983). MR 84g:20025
  • [GLS] D. Gorenstein, R. Lyons, & R. Solomon, The classification of the finite simple groups, Number 3, Math. Surveys and Monographs, vol. 40.3, Amer. Math. Soc. (1998). MR 98j:20011
  • [Gr] J. Grodal, Higher limits via subgroup complexes, Annals of Math. 155 (2002), 405-457.MR 2003g:55025
  • [Hf] G. Hoff, Cohomologies et extensions de categories, Math. Scand. 74 (1994), 191-207.MR 95k:18012
  • [HV] J. Hollender & R. Vogt, Modules of topological spaces, applications to homotopy limits and $E_\infty$ structures, Arch. Math. 59 (1992), 115-129.MR 93e:55015
  • [JM] S. Jackowski & J. McClure, Homotopy decomposition of classifying spaces via elementary abelian subgroups, Topology 31 (1992), 113-132.MR 92k:55026
  • [JMO] S. Jackowski, J. McClure, & B. Oliver, Homotopy classification of self-maps of $BG$ via $G$-actions, Annals of Math. 135 (1992), 183-270.MR 93e:55019a
  • [JMO2] S. Jackowski, J. McClure, & B. Oliver, Homotopy theory of classifying spaces of compact Lie groups, Algebraic topology and its applications, M.S.R.I. Publ. 27, Springer-Verlag (1994), 81-123. MR 95b:55001
  • [La] J. Lannes, Sur les espaces fonctionnels dont la source est le classifiant d'un $p$-groupe abélien élémentaire, Publ. Math. I.H.E.S. 75 (1992), 135-244.MR 93j:55019
  • [LS] J. Lannes & L. Schwartz, Sur la structure des $A$-modules instables injectifs, Topology 28 (1989), 153-169.MR 90h:55027
  • [LSS] M. Liebeck, J. Saxl, & G. Seitz, Subgroups of maximal rank in finite exceptional groups of Lie type, Proc. London Math. Soc. 65 (1992), 297-325.MR 93e:20026
  • [LW] M. Linckelmann & P. Webb, unpublished notes.
  • [Ol] B. Oliver, Higher limits via Steinberg representations, Comm. in Algebra 22 (1994), 1381-1393.MR 95b:18007
  • [Pu] L. Puig, Unpublished notes.
  • [Pu2] L. Puig, Full Frobenius systems and their localizing categories, preprint.
  • [Qu] D. Quillen, The spectrum of an equivariant cohomology ring: I, Annals of Math. 94 (1971), 549-572.MR 45:7743
  • [Qu2] D. Quillen, Higher algebraic $K$-theory I, Lecture notes in mathematics 341 (1973), 77-139.MR 49:2895
  • [Se] G. Segal, Categories and cohomology theories, Topology 13 (1974), 293-312.MR 50:5782
  • [Sol] R. Solomon, Finite groups with Sylow 2-subgroups of type $.3$, J. Algebra 28 (1974), 182-198.MR 49:9077
  • [Sw] R. Swan, The nontriviality of the restriction map in the cohomology of groups, Proc. Amer. Math. Soc. 11 (1960), 885-887. MR 23:A1370

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Additional Information

Carles Broto
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, E–08193 Bellaterra, Spain
Email: broto@mat.uab.es

Ran Levi
Affiliation: Department of Mathematical Sciences, University of Aberdeen, Meston Building 339, Aberdeen AB24 3UE, United Kingdom
Email: ran@maths.abdn.ac.uk

Bob Oliver
Affiliation: LAGA, Institut Galilée, Av. J-B Clément, 93430 Villetaneuse, France
Email: bob@math.univ-paris13.fr

DOI: https://doi.org/10.1090/S0894-0347-03-00434-X
Keywords: Classifying space, $p$-completion, finite groups, fusion.
Received by editor(s): August 3, 2001
Published electronically: July 21, 2003
Additional Notes: The first author is partially supported by MCYT grant BFM2001–2035
The second author is partially supported by EPSRC grant GR/M7831.
The third author is partially supported by UMR 7539 of the CNRS
All of the authors have been supported by EU grant HPRN-CT-1999-00119.
Article copyright: © Copyright 2003 American Mathematical Society

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