Rational isomorphisms of p-compact groups

Abstract

A rational isomorphism is a p-compact group homomorphism inducing an isomorphism on rational cohomology. Finite covering homomorphisms and nontrivial endomorphisms of simple p-compact groups are rational isomorphisms. It is shown that rational isomorphisms of p-compact groups restrict to admissible rational isomorphisms of the maximal tori and the classification of rational isomorphisms between connected p-compact groups is reduced to the simply connected case. The paper also contains a triviality criterion asserting that, on any connected p-compact group, only the trivial homomorphism induces the trivial map in rational cohomology. 0. Introduction The notion of a p-compact group was introduced by Dwyer and Wilkerson [9] as a homotopy theoretic candidate for a replacement of compact Lie groups. Subsequent investigation in [18] and [8] strengthened the candidacy in finding that much of the internal structure of compact Lie groups does seem to be present also in pcompact groups. The process of gathering support for p-compact groups continues here where the outlining idea is to translate Baum’s paper [3], describing local isomorphism systems of Lie groups, into the setting of p-compact groups. The rational isomorphisms of the title form the most prominent concept of this paper. However, for the sake of stressing the similarity with Lie groups, I shall in this introduction restrict myself to the the particular rational isomorphisms called finite covering homomorphisms: A finite covering homomorphism between p-compact groups is an epimorphism whose kernel is a finite p-group [Definition 2]. It was shown in [18] that any connected p-compact group X admits a finite covering homomorphism q : Y × S → X where Y is a simply connected p-compact group and S is a p-compact torus; the homomorphism q can even be chosen to be what is here called a special finite covering homomorphism. Locally isomorphic p-compact groups [Definition 3] are characterized by having isomorphic finite covering groups of this kind [Proposition 1.5]. As an example of how these concepts behave as our experience with Lie groups tells us they should, the following theorem — containing elements from Theorem 3.3 and Corollary 3.5 — is an almost mechanical translation of the basic lifting property 1991 Mathematics Subject Classification. 55P35, 55S37.