HOMOTOPY LIE GROUPS AND THEIR CLASSIFICATION
JESPER M. MØLLER
A p-compact group, where p is a prime number, is a p-complete space BX whose loop space
X = ΩBX has finite mod p singular cohomology. If G is a Lie group and π0 (G) is a finite p-group
then the p-completed classifying space of G is a p-compact group. The Sullivan spheres (BS 2n−1 )∧
p,
n|(p − 1), or, more generally, the Clark–Ewing spaces [2] are also examples of p-compact groups.
These homotopy Lie groups were defined and explored by W.G. Dwyer and C.W. Wilkerson in a
series of papers [5, 6, 7]. (See one of the survey articles articles [4, 8, 9] for a quick overview.)
They show that any p-compact group BX has a maximal torus BT → BX and a Weyl group W
which, for a connected p-compact group, is a p-adic reflection group acting on the maximal torus.
The normalizer BN → BX of the maximal torus is an extension
BT → BN → BW
of the maximal torus by the Weyl group. (Strictly speaking, BN is in general not a p-compact
group as its fundamental group may not be a finite p-group; instead, BN is an example of what is
called an extended p-compact torus.)
As pointed out by Curtis, Wiederhold, and Williams [3], compact connected Lie groups are
determined by the normalizer of a maximal torus. By analogy, it has been conjectured that the
same statement might also hold for p-compact groups. This classification conjecture has been
verified for odd primes [11, 12, 1], and for p = 2 there is now a partial result [10].
The talk will be centered around these headlines:
• A general introduction to the theory of p-compact groups.
• The classification of p-compact groups at odd primes and some consequences.
• Towards the classification of 2-compact groups and some possible consequences.
References
[1] K. Andersen, J. Grodal, J. Møller, and A. Viruel, The classification of p-compact groups for p odd, Ann of
Math (to appear).
[2] Allan Clark and John Ewing, The realization of polynomial algebras as cohomology rings, Pacific J. Math. 50
(1974), 425–434. MR 51 #4221
[3] Morton Curtis, Alan Wiederhold, and Bruce Williams, Normalizers of maximal tori, Localization in group
theory and homotopy theory, and related topics (Sympos., Battelle Seattle Res. Center, Seattle, Wash., 1974),
Springer, Berlin, 1974, pp. 31–47. Lecture Notes in Math., Vol. 418. MR 51 #13131
[4] W. G. Dwyer, Lie groups and p-compact groups, Proceedings of the International Congress of Mathematicians,
Vol. II (Berlin, 1998), no. Extra Vol. II, 1998, pp. 433–442 (electronic). MR 99h:55025
[5] W. G. Dwyer and C. W. Wilkerson, Homotopy fixed-point methods for Lie groups and finite loop spaces, Ann.
of Math. (2) 139 (1994), no. 2, 395–442. MR 95e:55019
[6]
, The center of a p-compact group, The Čech centennial (Boston, MA, 1993) (Providence, RI), Amer.
Math. Soc., 1995, pp. 119–157. MR 96a:55024
[7]
, Product splittings for p-compact groups, Fund. Math. 147 (1995), no. 3, 279–300. MR 96h:55005
[8] Jean Lannes, Théorie homotopique des groupes de Lie (d’après W. G. Dwyer et C. W. Wilkerson), Astérisque
(1995), no. 227, Exp. No. 776, 3, 21–45, Séminaire Bourbaki, Vol. 1993/94. MR 96b:55017
[9] Jesper M. Møller, Homotopy Lie groups, http://www.math.ku.dk/~moller/preprints/lillenotes.pdf.
[10]
, N -determined 2-compact groups, http://www.math.ku.dk/~moller/preprints/ndet2cgs.pdf.
[11]
, Deterministic p-compact groups, Stable and unstable homotopy (Toronto, ON, 1996), Fields Inst.
Commun., vol. 19, Amer. Math. Soc., Providence, RI, 1998, pp. 255–278. MR 99b:55012
[12]
, N -determined p-compact groups, Fund. Math. 173 (2002), no. 3, 201–300. MR 1 925 483
Matematisk Institut, Universitetsparken 5, DK–2100 København
E-mail address: moller@math.ku.dk
URL: http://www.math.ku.dk/~moller
Date: 15th April 2005.
1