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started a stub, to satisfy links at Dwyer-Wilkerson space
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and added in a section “Properties – Classification” this bit, adapted from Grodal’s AlgTop message today:
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 $\mathbb{Z}$.
There are a few equivalent definitions, even in Dwyer-Wilkerson. Another definition is a space $X$ such that its delooping $BX$ is $p$-local and its Z/pZ-cohomology ring is finitely generated as a Z/pZ-module.
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