P-compact group
In mathematics, in particular algebraic topology, a "p"-compact group is a homotopical version of a compact Lie group, but with all the local structure concentrated at a single prime "p". This concept was introduced in , making precise earlier notions of a mod p finite loop space. A p-compact group has many Lie-like properties like maximal tori and Weyl groups, which are defined purely homotopically in terms of the classifying space, but with the important difference that the Weyl group, rather than being a finite reflection group over the integers, is now a finite "p"-adic reflection  group. They admit a classification in terms of root data, which mirrors the classification of compact Lie groups, but with the integers replaced by the "p"-adic integers. Definition. A "p"-compact group is a pointed space "BG", with is local with respect to mod "p" homology, and such the pointed loop space "G = ΩBG" has finite mod "p" homology. One sometimes also refer to the "p"-compact group by "G", but then one needs to keep in mind that the loop space structure is part of the data (which then allows one to recover "BG"). A "p"-compact group is said to  be connected if G is a connected space (in general the group of components of G will be a finite p-group). The rank of a "p"-compact group is the rank of its maximal torus. Classification. The classification of "p"-compact groups from states that there is a 1-1 correspondence between connected "p"-compact groups, up to homotopy equivalence, and root data over the "p"-adic integers, up to isomorphism. This is analogous to the classical classification of connected compact Lie groups, with the "p"-adic integers replacing the rational integers. It follows from the classification that any "p"-compact group can be written  as "BG = BH × BK" where "BH" is the "p"-completion of a compact connected Lie group and BK is finite direct product of simple "exotic" p-compact groups i.e., simple p-compact groups whose Weyl group group is not a ℤ-reflection groups. Simple exotic p-compact groups are again in 1-1-correspondence with irreducible complex reflection groups whose character field can be embedded in ℚ, but is not ℚ. For instance, when "p=2" this implies that every connected 2-compact group can be written "BG = BH × BDI(4)", where BH is the 2-completion of the classifying  space of a connected compact Lie group, and BDI(4) denotes s copies of the "Dwyer-Wilkerson 2-compact group" BDI(4) of rank 3, constructed in with Weyl group corresponding to group number 24 in the Shepard-Todd enumeration of complex reflection groups. For "p=3" a similar statement holds but the new exotic 3-compact group is now group number 12 on the Shepard-Todd list, of rank 2. For primes greater than 3, family 2 on the Shepard-Todd list will contain infinitely many exotic p-compact groups. Some consequences of the classification. A finite loop space is a pointed  space BG such that the loop space ΩBG is homotopy equivalent to a finite CW-complex. The classification of connected p-compact groups implies a "classification of connected finite loop spaces": Given a connected p-compact group for each prime, all with the same rational type, there is an explicit double coset space of possible connected finite loop spaces with p-completion the give p-compact groups. As connected p-compact groups are classified combinatorially, this implies a classification of connected loop spaces as well. Using the classification, one can identify the compact Lie groups inside finite loop spaces, giving  a "homotopical characterisation of compact connected Lie groups": They are exactly those finite loop spaces that admit an integral maximal torus; this was the so-called "maximal torus conjecture". (See and .) The classification also implies a classification of which graded polynomial rings can occur as the cohomology ring of a space, the so-called "Steenrod problem". (See .)