THE CENTER OF A p{COMPACT GROUP

Compact Lie groups appear frequently in algebraic topology, but they are relatively rigid analytic objects, and partially for that reason are a challenge to understand homotopically. Since H. Hopf (and H-spaces) topologists have aspired to escape the analytic straitjacket by finding some homotopy theoretic concept which would capture enough of the idea of “compact Lie group” to lead to rich and interesting structure theorems. In [12] we came up with our own candidate, the notion of p-compact group, and studied these objects to the extent of constructing maximal tori, Weyl groups, etc. In this paper we continue the study by looking at the idea of the “center” of a p-compact group and showing that two very different ways of defining the center are equivalent. This leads for instance to a reproof and generalization of a theorem from [15] about the identity component of the space of self homotopy equivalences of BG (G compact Lie). Along the way we find various familiar-looking elements of internal structure in a p-compact group X , enumerate the X ’s which are abelian in the appropriate sense, and construct what might be called the “adjoint form” of X . Before describing in more detail the main results we are aiming at, we have to introduce some ideas from [12]. A loop space X is by definition a triple (X ,BX , e) in which X is a space, BX is a connected pointed space (called the classifying space of X ), and e : X → ΩBX is a homotopy equivalence from X to the space ΩBX of based loops in BX . A p-compact group is a loop space X such that (1) X is Fp-finite (in the sense that H∗(X ; Fp) is finite dimensional), (2) π0X is a finite p-group, and (3) πiX (i ≥ 1) is a finitely generated module over the ring Zp of p-adic integers. A homomorphism f : X → Y between loop spaces is a pointed map Bf : BX → BY; if X and Y are p-compact groups the homomorphism f is said to be a monomorphism if the homotopy fibre Y/f(X ) of Bf is Fp-finite (in this case, if the homomorphism f is understood, we will refer to X as a subgroup of Y). If f : X → Y is a homomorphism of loop spaces, the centralizer of f(X ) in Y, denoted CY(f(X )), is the loop space Ω Map(BX ,BY)Bf . Here Map(BX ,BY)Bf is the component containing Bf of the space of (unpointed) maps