p-compact group

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.

• The p-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 the p-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 connected 2-compact group is either the 2-completion of SU(2) or SO(3). A rank 1 connected p-compact group, for p odd, is a "Sullivan sphere", i.e., the p-completion of a 2n-1-sphere S^2n-1, where n divides p − 1. These spheres turn out to have a unique loop space structure. They were first constructed by Dennis Sullivan in his 1970 MIT notes. (The Weyl group is a cyclic group of order n, acting on ${\displaystyle \mathbb {Z} _{p}}$ via an nth root of unity.)

• Generalizing the rank 1 case, any finite complex reflection group ${\displaystyle W\leq GL_{r}(\mathbb {C} )}$ can be realized as the Weyl group of a p-compact group for infinitely many primes, with the primes being determined by whether W and be conjugated into ${\displaystyle GL_{r}(\mathbb {Z} _{p})}$ or not, with some embedding of ${\displaystyle \mathbb {Z} _{p}}$ in ${\displaystyle \mathbb {C} }$. The construction of a p-compact group with this Weyl group is then relatively straightforward for large primes where p does not divide the order of W (carried out already in Clark & Ewing (1974) using the Chevalley–Shephard–Todd theorem), but requires more sophisticated methods for the "modular primes" p that divide the order of W.

The classification of p-compact groups from Andersen & Grodal (2009) 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 ${\displaystyle \mathbb {Z} }$-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, when p=2 this implies that every connected 2-compact group can be written BG = 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 in Dwyer & Wilkerson (1993) 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.

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 Andersen & Grodal (2009) and Grodal (2010).)

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 Andersen & Grodal (2008).)

• Andersen, K.K.S.; Grodal, J.; Møller, J.; Viruel, A. (2008), "The classification of p-compact groups for p odd", Ann. of Math. (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", J. Amer. Math. Soc., 22 (2): 387–436, arXiv:math/0611437, 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", J. Topol., 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 J. Math., 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", Ann. of Math. (2), 139: 395–442, doi:10.2307/2946585, JSTOR 2946585
• Dwyer, W.G; Wilkerson, C.W (1993), "A new finite loop space at the prime 2", J. Amer. Math. Soc., 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, arXiv:1003.4010
• Homotopy Lie Groups: A Survey (PDF)
• Homotopy Lie Groups and Their Classification (PDF)