p-compact group

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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^[1]. Subsequently the name homotopy Lie group has also been used.

Contents 1 Examples 2 Classification 3 References 4 Notes

[edit] 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

2n − 1,

if n divides p − 1.

[edit] 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.

[edit] References

• Homotopy Lie Groups: A Survey (PDF)
• Homotopy Lie Groups and Their Classification (PDF)

[edit] Notes

1. ^ 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.