Geometry & Topology

Automorphisms of $p$–compact groups and their root data

Kasper K S Andersen and Jesper Grodal

Full-text: Open access


We construct a model for the space of automorphisms of a connected p–compact group in terms of the space of automorphisms of its maximal torus normalizer and its root datum. As a consequence we show that any homomorphism to the outer automorphism group of a p–compact group can be lifted to a group action, analogous to a classical theorem of de Siebenthal for compact Lie groups. The model of this paper is used in a crucial way in our paper ‘The classification of 2-compact groups’ [arXiv:math.AT/0611437], where we prove the conjectured classification of 2–compact groups and determine their automorphism spaces.

Article information

Geom. Topol., Volume 12, Number 3 (2008), 1427-1460.

Received: 11 January 2007
Revised: 1 April 2008
Accepted: 30 November 2007
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55R35: Classifying spaces of groups and $H$-spaces
Secondary: 20G99: None of the above, but in this section 22E15: General properties and structure of real Lie groups 55P35: Loop spaces

$p$-compact group root datum maximal torus normalizer


Andersen, Kasper K S; Grodal, Jesper. Automorphisms of $p$–compact groups and their root data. Geom. Topol. 12 (2008), no. 3, 1427--1460. doi:10.2140/gt.2008.12.1427.

Export citation


  • K K S Andersen, The normalizer splitting conjecture for $p$–compact groups, from: “Algebraic Topology. Proceedings of the 4th Conference held in Kazimierz Dolny, June 12–19, 1997”, (W Dwyer, S Jackowski, editors), Fund. Math. 161 (1999) 1–16
  • K K S Andersen, J Grodal, The classification of $2$–compact groups
  • K K S Andersen, J Grodal, J M Møller, A Viruel, The classification of $p$–compact groups for $p$ odd, Ann. of Math. $(2)$ 167 (2008) 95–210
  • T Bauer, N Kitchloo, D Notbohm, E K Pedersen, Finite loop spaces are manifolds, Acta Math. 192 (2004) 5–31
  • G W Bell, On the cohomology of the finite special linear groups. I, II, J. Algebra 54 (1978) 216–238, 239–259
  • D Benson, Modular representation theory: new trends and methods, Lecture Notes in Math. 1081, Springer, Berlin (1984)
  • A Borel, Linear algebraic groups, second edition, Graduate Texts in Math. 126, Springer, New York (1991)
  • N Bourbaki, Éléments de mathématique: groupes et algèbres de Lie. Chapitre 9. Groupes de Lie réels compacts, Masson, Paris (1982)
  • A K Bousfield, D M Kan, Homotopy limits, completions and localizations, Lecture Notes in Math. 304, Springer, Berlin (1972)
  • C Broto, J M Møller, Chevalley $p$–local finite groups, Algebr. Geom. Topol. 7 (2007) 1809–1919
  • K S Brown, Cohomology of groups, Graduate Texts in Math. 87, Springer, New York (1994) Corrected reprint of the 1982 original
  • A Clark, J Ewing, The realization of polynomial algebras as cohomology rings, Pacific J. Math. 50 (1974) 425–434
  • M Curtis, A Wiederhold, B Williams, Normalizers of maximal tori, from: “Localization in group theory and homotopy theory, and related topics (Sympos., Battelle Seattle Res. Center, Seattle, Wash., 1974)”, Lecture Notes in Math. 418, Springer, Berlin (1974) 31–47
  • W G Dwyer, Lie groups and $p$–compact groups, from: “Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998)”, Extra Vol. II (1998) 433–442
  • W G Dwyer, H R Miller, C W Wilkerson, The homotopic uniqueness of $BS\sp 3$, from: “Algebraic topology, Barcelona, 1986”, Lecture Notes in Math. 1298, Springer, Berlin (1987) 90–105
  • W G Dwyer, C W Wilkerson, Homotopy fixed-point methods for Lie groups and finite loop spaces, Ann. of Math. $(2)$ 139 (1994) 395–442
  • W G Dwyer, C W Wilkerson, The center of a $p$–compact group, from: “The Čech centennial (Boston, 1993)”, Contemp. Math. 181, Amer. Math. Soc. (1995) 119–157
  • W G Dwyer, C W Wilkerson, Product splittings for $p$–compact groups, Fund. Math. 147 (1995) 279–300
  • W G Dwyer, C W Wilkerson, Normalizers of tori, Geom. Topol. 9 (2005) 1337–1380
  • P G Goerss, J F Jardine, Simplicial homotopy theory, Progress in Math. 174, Birkhäuser Verlag, Basel (1999)
  • J-F Hämmerli, The outer automorphism group of normalizers of maximal tori in connected compact Lie groups, J. Lie Theory 12 (2002) 357–368
  • J-F Hämmerli, M Matthey, U Suter, Automorphisms of normalizers of maximal tori and first cohomology of Weyl groups, J. Lie Theory 14 (2004) 583–617
  • J E Humphreys, Linear algebraic groups, Graduate Texts in Math. 21, Springer, New York (1975)
  • J E Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Math. 9, Springer, New York (1978) Second printing, revised
  • J E Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Math. 29, Cambridge University Press (1990)
  • S Jackowski, J McClure, B Oliver, Homotopy classification of self-maps of $BG$ via $G$–actions. II, Ann. of Math. $(2)$ 135 (1992) 227–270
  • S Jackowski, J McClure, B Oliver, Self-homotopy equivalences of classifying spaces of compact connected Lie groups, Fund. Math. 147 (1995) 99–126
  • W Jones, B Parshall, On the $1$–cohomology of finite groups of Lie type, from: “Proceedings of the Conference on Finite Groups (Univ. Utah, Park City, Utah, 1975)”, Academic Press, New York (1976) 313–328
  • M Matthey, Normalizers of maximal tori and cohomology of Weyl groups, preprint (2002) available online at
  • D Notbohm, On the “classifying space” functor for compact Lie groups, J. London Math. Soc. $(2)$ 52 (1995) 185–198
  • A Osse, $\lambda$–structures and representation rings of compact connected Lie groups, J. Pure Appl. Algebra 121 (1997) 69–93
  • C H Sah, Cohomology of split group extensions, J. Algebra 29 (1974) 255–302
  • J de Siebenthal, Sur les groupes de Lie compacts non connexes, Comment. Math. Helv. 31 (1956) 41–89
  • T A Springer, Linear algebraic groups, second edition, Progress in Math. 9, Birkhäuser, Boston (1998)
  • J Tits, Normalisateurs de tores. I. Groupes de Coxeter étendus, J. Algebra 4 (1966) 96–116