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 [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.] . Subsequently the name homotopy Lie group has also been used.

Examples

Examples include the p-completion of a compact and connected Lie group, and the Sullivan spheres, i.e. the "p"-completion of a sphere of dimension

:2"n" − 1,

if "n" divides "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.

References

* [http://www.math.ku.dk/~moller/preprints/lillenotes.pdf "Homotopy Lie Groups: A Survey" (PDF)]
* [http://www.math.uio.no/~stolen/moeller.pdf "Homotopy Lie Groups and Their Classification" (PDF)]

Notes