The fundamental group of a p -compact group

Abstract

The notion of p-compact group [10] is a homotopy theoretic version of the geometric or analytic notion of compact Lie group, although the homotopy theory differs from the geometry is that there are parallel theories of p-compact groups, one for each prime number p. A key feature of the theory of compact Lie groups is the relationship between centers and fundamental groups; these play off against one another, at least in the semisimple case, in that the center of the simply connected form is the fundamental group of the adjoint form. There are explicit ways to compute the center or fundamental group of a compact Lie group in terms of the normalizer of the maximal torus [1, 5.47]. For some time there has in fact been a corresponding formula for the center of a p-compact group [11, 7.5], but in general the fundamental group has eluded analysis. The purpose of the present paper is to remedy this deficit. For any space Y , we let H Zp i (Y ) denotes lim nHi(Y ;Z/p ). Suppose that X is a connected p-compact group, with maximal torus T and torus normalizer NT [10, §8]. It is known that the map π1(T ) → π1(X) is surjective [12, 6.11] [21, 5.6], or equivalently that the map H Zp 2 (BT ) → H Zp 2 (BX) is surjective. We prove the following statement.