nLab p-compact group

Idea

The $p$-compact groups (Dwyer-Wilkerson 94) for some prime number $p$ are a class of ∞-groups that shares crucial properties with the class of compact Lie groups, without actually being compact Lie groups. Hence they are also called homotopy Lie groups (Møller 95).

Properties

Classification

The classification of $p$-compact groups states that there is a bijection between isomorphism classes of connected p-compact groups, and isomorphism classes of root data over the p-adic integers (as conjectured by Clarence Wilkerson and others, in various forms, since the early days of the theory).

This is completely analogous to the classification of connected compact Lie groups, under replacing the integers $\mathbb{Z}$ by the p-adic integers $\mathbb{Z}_p$.

Specializing to $p=2$ one gets as a corollary that any classifying space $B X$ of a connected 2-compact group $X$ splits as

$$B X \cong B G \times \big(B DI(4)\big)^s$$

the Cartesian product of the 2-completion of the classifying space of the compact Lie group $G$, and $s$ copies of the Dwyer-Wilkerson space $B DI(4)$ for some $s$.

$DI(4) =$ G3 corresponds to the finite $\mathbb{Z}_2$-reflection group which is number 24 on the Shepard-Todd list. It is the only irreducible finite complex reflection group which is realizable over $\mathbb{Z}_2$ but not $\mathbb{Z}$.

(Andersen-Grodal 06, see Grodal 10)

Examples

• Let $G$ be any compact Lie group whose component group $\pi_0(G)$ is a $p$-group. Define $B \hat{G} = (B G)_p$. Then $\hat{G}$ is a $p$-compact group.

• (Sullivan) The $\mathbb{F}_p$-local sphere $(S^{n-1})_p$, where $n \gt 2$ is an integer dividing $p-1$.

• Dwyer-Wilkerson H-space

References

Due to

• William Dwyer, Clarence Wilkerson, Homotopy Fixed-point Methods for Lie Groups and Finite Loop Spaces, Annals of Mathematics Second Series, Vol. 139, No. 2 (Mar., 1994), pp. 395-442 (jstor:2946585)

Classification:

• Kasper Andersen, Jesper Grodal, The classification of 2-compact groups, J. Amer. Math. Soc. 22 (2009), 387-436 (arXiv:math/0611437)

Review:

• Jesper Møller, Homotopy Lie groups, Bull. Amer. Math. Soc. (N.S.) 32 (1995) 413-428 (arXiv:math/9510218)

• Jesper Grodal, The Classification of $p$–Compact Groups and Homotopical Group Theory, Proceedings of the International Congress of Mathematicians, Hyderabad 2010 (arXiv:1003.4010, pdf, pdf)

See also

• Wikipedia, P-compact group