p-compact group

In mathematics, in particular algebraic topology, a p-compact group is (roughly speaking) a space that is a homotopical version of a compact Lie group, but with all the structure concentrated at a single prime p. This concept was introduced by Dwyer and Wilkerson in Dwyer & Wilkerson (1994), making precise earlier notions of a mod p finite loop space. A p-compact group has many Lie-like properties like maximal tori and Weyl groups, which are defined purely homotopically, but with the important difference that the Weyl group is a finite p-adic reflectection group, rather than a finite reflection group over the integers.

Definition[]

A p-compact group is a pointed space BG, with is local with respect to mod p homology, and such the pointed loop space G = ΩBG has finite mod p homology. One sometimes also refer to the p-compact group by G, but then one needs to keep in mind that the loop space structure is part of the data (which allows one to recover BG). A p-compact group is said to be connected if G is a connected space (in general the group of components of G will be a finite p-group). The rank of a p-compact group is the rank of its maximal torus.

Examples[]

Examples include the of a compact and connected Lie group, and more generally the p-completion of a connected finite loop space.

A rank one connected 2-compact group is either the 2-completion of SU(2) or SO(3). If p is odd, the rank one p-compact groups are the , i.e. the p-completion of a 2n-1-sphere S^2n-1, for those n that divide p − 1.

Classification[]

The classification of p-compact groups states that there is a 1-1 correspondence between connected p-compact groups, and root data over the p-adic integers. This is analogous to the classical classification of connected compact Lie groups, with the p-adic integers replacing the rational integers.

For instance at the prime 2 this implies that every connected 2-compact group can be written BG = BH × BDI(4)^s, where BH is the 2-completion of the classifying space of a connected compact Lie group, and BDI(4)^s denotes s copies of the "Dwyer-Wilkerson 2-compact group" of rank 3.

References[]

K.K.S. Andersen, J. Grodal, J. Møller, A. Viruel: "The classification of p-compact groups for p odd". Ann. of Math. (2) 167 (2008), no. 1, 95–210.
K.K.S. Andersen and J. Grodal: "The classification of 2-compact groups". J. Amer. Math. Soc. 22 (2009), no. 2, 387–436.
Dwyer, W.G; Wilkerson, C.W (1994), "Homotopy fixed-point methods for Lie groups and finite loop spaces", Ann. of Math. (2), 139: 395–442
W. G. Dwyer and C. W. Wilkerson, A new finite loop space at the prime 2, J. Amer. Math. Soc. 6 (1993), no. 1, 37–64.
J. Grodal "The classification of p-compact groups and homotopical group theory" (ICM 2010 Survey)
Homotopy Lie Groups: A Survey (PDF)
Homotopy Lie Groups and Their Classification (PDF)

Notes[]

This topology-related article is a stub. You can help Wikipedia by .