A major open problem in the homotopy theory of Lie groups is the classification of 2-compact groups, analogous to the classification of compact Lie groups. The corresponding conjecture for p odd has been settled in [2], after much work by many people. (See also Dwyer's ICM address [5] for earlier history.)
Conjecture 1 There is a 1-1 correspondence between
Furthermore
given by associating to (X,BX,e) the root datum (WX,p1(T),p1([T\tilde])), where WX and T is the Weyl group and maximal torus of X, and [T\tilde] is the maximal torus of the universal cover [X\tilde].
connected 2-compact groups « Z2-root data (W,L,L0)
p0(Aut(BX)) @ Out((WX,L,L0)) = { j Î NGL(L)(W)/W | j(L0) = L0}
It is easy to enumerate all Z2-root data, in a way analogous to the classification of classical root data (over Z or R). Hence the right hand side is to be considered completely understood.
Part of the trouble in proving this conjecture stems from the fact that there is no known way of constructing a (classical) compact Lie group from its root datum, without going via the Lie algebra. Since the Lie algebra is not a homotopy theoretic object this tool is not available for p-compact groups. The method of proof employed in [2] for odd primes uses obstruction theory and shows promise of being extendable to p=2, although much work remains to be done....
Results about p-compact groups can often be translated into results about finite loop spaces via Sullivan's arithmetic square. For instance this has been used to show in [3] that every finite loop space is homotopy equivalent to a smooth parallelizable manifold. It has also been used in [1] to produce an exotic finite loop space which is not rationally homotopy equivalent to any compact Lie group, resolving an old conjecture in the negative.
Problem 2 Use geometric methods to concretely describe the manifolds constructed via the theory of p-compact groups. How many manifolds are there in each homotopy class? What can be said about them? Do they admit an explicit description?
The p-local finite group of a group G roughly carries information about the Sylow p-subgroup S as well as information about which subgroups in it are conjugate or ``fused'' in G. Not all p-local finite groups come from finite groups, but the exceptions seem to be rather limited. Indeed in ``most'' cases it seems that the p-local finite group determines a unique ambient finite group.
More concretely, a p-local finite group is a certain category L, where the objects should be thought of as p-subgroups and the morphisms as elements conjugating the one subgroup into the other. (See [4] for basic definitions.) To a p-local finite group L we can associate a space |L|, the nerve of the category L. One can in fact recover L from the homotopy type of |L|.
Problem 3
Classify simple p-local finite groups and compare the result to the classification of finite simple groups.