nLab
p-compact group

Contents

Context

Group Theory

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

The pp-compact groups (Dwyer-Wilkerson 94) for some prime number pp 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 pp-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 p\mathbb{Z}_p.

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

BXBG×(BDI(4)) s 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 GG, and ss copies of the Dwyer-Wilkerson space BDI(4)B DI(4) for some ss.

DI(4)=DI(4) = G3 corresponds to the finite 2\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 2\mathbb{Z}_2 but not \mathbb{Z}.

(Andersen-Grodal 06, see Grodal 10)

Examples

  • Let GG be any compact Lie group whose component group π 0(G)\pi_0(G) is a pp-group. Define BG^=(BG) pB \hat{G} = (B G)_p. Then G^\hat{G} is a pp-compact group.

  • (Sullivan) The 𝔽 p\mathbb{F}_p-local sphere (S n1) p(S^{n-1})_p, where n>2n \gt 2 is an integer dividing p1p-1.

  • Dwyer-Wilkerson H-space

References

Due to

Classification:

Review:

See also

Last revised on August 28, 2019 at 12:47:04. See the history of this page for a list of all contributions to it.