Inmathematics, in particularalgebraic topology, ap-compact group is ahomotopical version of acompactLie group, but with all the local structure concentrated at a singleprimep. This concept was introduced inDwyer & Wilkerson (1994), making precise earlier notions of a mod p finite loop space. A p-compact group has many Lie-like properties likemaximal tori andWeyl groups, which are defined purely homotopically in terms of the classifying space, but with the important difference that theWeyl group, rather than being a finitereflection group over the integers, is now a finitep-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 thep-adic integers.

Ap-compact group is a pointed spaceBG, which is local with respect to modphomology, and such the pointedloop spaceG = ΩBG has finite modp homology. One sometimes also refer to thep-compact group byG, but then one needs to keep in mind that the loop space structure is part of the data (which then allows one to recoverBG).

Ap-compact group is said to beconnected if G is aconnected space (in general the group of components of G will be a finite p-group). Therank of ap-compact group is the rank of its maximal torus.

• Thep-completion, in the sense of homotopy theory, of (the classifying space of) a compact connected Lie group defines a connected p-compact group. (The Weyl group is just its ordinary Weyl group, now viewed as a p-adic reflection group by tensoring the coweight lattice by${\displaystyle {\mathbb{Z}}_p}$.)

• More generally thep-completion of a connected finite loop space defines a p-compact group. (Here the Weyl will be a${\displaystyle {\mathbb{Z}}_p}$-reflection group that may not come from a${\displaystyle \mathbb{Z}}$-reflection group.)

• A rank 1 connected2-compact group is either the2-completion ofSU(2) orSO(3). A rank 1 connected p-compact group, for p odd, is a "Sullivan sphere", i.e., thep-completion of a2n-1-sphere S^2n-1, wheren dividesp −1. These spheres turn out to have a unique loop space structure. They were first constructed byDennis Sullivan in his 1970 MIT notes. (The Weyl group is a cyclic group of ordern, acting on${\displaystyle {\mathbb{Z}}_p}$ via annth root of unity.)

• Generalizing the rank 1 case, any finitecomplex reflection group${\displaystyle W\le G{L}_r(\mathbb{C})}$ can be realized as the Weyl group of ap-compact group for infinitely many primes, with the primes being determined by whetherW and be conjugated into${\displaystyle G{L}_r({\mathbb{Z}}_p)}$ or not, with some embedding of${\displaystyle {\mathbb{Z}}_p}$ in${\displaystyle \mathbb{C}}$. The construction of ap-compact group with this Weyl group is then relatively straightforward for large primes wherep does not divide the order ofW (carried out already inClark & Ewing (1974) using theChevalley–Shephard–Todd theorem), but requires more sophisticated methods for the "modular primes"p that divide the order ofW.

The classification ofp-compact groups fromAndersen & Grodal (2009) states that there is a 1-1 correspondence between connectedp-compact groups, up to homotopy equivalence, androot data over thep-adic integers, up to isomorphism. This is analogous to the classical classification of connected compact Lie groups, with thep-adic integers replacing therational integers.

It follows from the classification that anyp-compact group can be written asBG = BH × BK whereBH is thep-completion of a compact connected Lie group and BK is finite direct product of simpleexotic p-compact groups i.e., simple p-compact groups whose Weyl group group is not a${\displaystyle \mathbb{Z}}$-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${\displaystyle {\mathbb{Q}}_p}$, but is not${\displaystyle \mathbb{Q}}$.

For instance, whenp=2 this implies that every connected 2-compact group can be writtenBG = BH × BDI(4)^s, where BH is the 2-completion of the classifying space of a connected compact Lie group, and BDI(4)^s denotes s copies of the "Dwyer-Wilkerson 2-compact group" BDI(4) of rank 3, constructed inDwyer & Wilkerson (1993) with Weyl group corresponding to group number 24 in theShepard-Todd enumeration ofcomplex reflection groups. Forp=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.

Afinite 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 aclassification 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 ahomotopical characterisation of compact connected Lie groups: They are exactly those finite loop spaces that admit an integral maximal torus; this was the so-calledmaximal torus conjecture. (SeeAndersen & Grodal (2009) andGrodal (2010).)

The classification also implies a classification of which graded polynomial rings can occur as the cohomology ring of a space, the so-calledSteenrod problem. (SeeAndersen & Grodal (2008).)

• Andersen, K.K.S.; Grodal, J.; Møller, J.; Viruel, A. (2008), "The classification of p-compact groups for p odd",Annals of Mathematics, Series 2,167:95–210,arXiv:math/0302346,doi:10.4007/annals.2008.167.95,S2CID 119168267
• Andersen, K.K.S.; Grodal, J. (2009), "The classification of 2-compact groups",Journal of the American Mathematical Society,22 (2):387–436,arXiv:math/0611437,Bibcode:2009JAMS...22..387A,doi:10.1090/S0894-0347-08-00623-1,S2CID 17542829
• Andersen, K.K.S.; Grodal, J. (2008),"The Steenrod problem of realizing polynomial cohomology rings",Journal of Topology,1 (4):747–760,arXiv:0704.4002,doi:10.1112/jtopol/jtn021,S2CID 1583621
• Clark, A.; Ewing, J. (1974),"The realization of polynomial algebras as cohomology rings",Pacific Journal of Mathematics,50 (2):425–434,doi:10.2140/pjm.1974.50.425
• Dwyer, W.G; Wilkerson, C.W (1994), "Homotopy fixed-point methods for Lie groups and finite loop spaces",Annals of Mathematics, Series 2,139 (2):395–442,doi:10.2307/2946585,JSTOR 2946585
• Dwyer, W.G; Wilkerson, C.W (1993), "A new finite loop space at the prime 2",Journal of the American Mathematical Society,6:37–64,doi:10.1090/S0894-0347-1993-1161306-9
• Grodal, J. (2010), "The classification of p-compact groups and homotopical group theory",Proceedings of the International Congress of Mathematicians 2010:973–1001,arXiv:1003.4010
• Homotopy Lie Groups: A Survey (PDF)
• Homotopy Lie Groups and Their Classification (PDF)

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