Lie Groups and p-Compact Groups

A p-compact group is the homotopical ghost of a compact Lie group; it is the residue that remains after the geometry and algebra have been stripped away. This paper sketches the theory of p-compact groups, with the intention of illustrating the fact that many classical structural properties of compact Lie groups depend only on homotopy theoretic considerations. 1 From compact Lie groups to p-compact groups The concept of p-compact group is the culmination of a series of attempts, stretching over a period of decades, to isolate the key homotopical characteristics of compact Lie groups. It has been something of a problem, as it turns out, to determine exactly what these characteristics are. Probably the first ideas along these lines were due to Hopf [10] and Serre [31]. 1.1. Definition. A finite H-space is a pair (X,m), where X is a finite CW– complex with basepoint ∗ and m : X × X → X is a multiplication map with respect to which ∗ functions, up to homotopy, as a two-sided unit. The notion of compactness in captured here in the requirement that X be a finite CW–complex. To obtain a structure a little closer to group theory, one might also ask that the multiplication on X be associative up to homotopy. Finite H-spaces have been studied extensively; see [18] and its bibliography. Most of the results deal with homological issues. There are a few general classification theorems, notably Hubbuck’s theorem [11] that any path-connected homotopy commutative finite H-space is equivalent at the prime 2 to a torus; this is a more or less satisfying analog of the classical result that any connected abelian compact Lie group is a torus. Experience shows, though, that there is little hope of understanding the totality of all finiteH-spaces, or even all homotopy associative ones, on anything like the level of detail that is achieved in the theory of compact Lie groups. The problem is that there are too many finite H-spaces; the structure is too lax. Stasheff pointed out one aspect of this laxity [32] that is particularly striking when it comes to looking at finite H-spaces as models for group theory. He discovered a whole hierarchy of generalized associativity conditions, all of a homotopy Documenta Mathematica · Extra Volume ICM 1998 · 1–1000