Compact group actions and maps into aspherical manifolds

Abstract

A WELL-KNOWN theorem of Cartan and Hadamard states that a complete Riemannian n-manifold with everywhere nonpositive sectional curvature has a universal covering that is diffeomorphic to Euclidean n-space ([23, p. 184). This is one of many motivations for interest in spaces with contractible universal coverings; such spaces are called aspherical. Over the past dozen years considerable information has been obtained regarding group actions on such manifolds (compare[9-12]), and recently analogous questions for manifolds admitting suitably nontrivial maps into certain aspherical manifolds have been considered. In particular, the second named author has considered group actions on closed n-manifolds that map to the n-torus by a degree one map[22,23] and Schoen and Yau have considered smooth or real analytic actions on n-manifolds that- map to closed Riemannian manifolds of nonpositive curvature [21]. (Compare also [29]). In this paper, we shall prove results that overlap or contain the results of Borel-Conner-Raymond on aspherical manifolds, the results of [22], and the theorems of Schoen and Yau on group actions. One class,of results deals with manifolds that map onto a closed aspherical manifold by a map of nonzero degree; this class of manifolds contains all closed aspherical manifolds and in fact all connected sums of closed aspherical manifolds with other manifolds. The results of Borel, Conner and Raymond generalize completely; the only connected compact Lie groups that can act are tori, and all such actions are injective in the sense of [9]. In fact, if f : A4 + N is the map from M to the closed aspherical manifold N and i : T + M is an orbit map, then the composite is injective. Furthermore, if the map f has degree one, then (i) a finite group G acting on M with fixed points induces an injection from G to Aut(n,(M, WI,,)), where m. is a fixed point, (ii) if rr,(M, mo) is centerless, then there is an injection of G into the outer automorphism group of r,(M, mo) even if G has no fixed points. The results of [21] also indicate that one has strong restrictions on the differential symmetry of M, even if dim N > dim M, so long as f maps the fundamental class of M sufficiently nontrivially and N has nonpositive curvature. Our results show that similar restrictions also hold on the topological symmetry of M. Technically speaking, the central ideas of the proofs are (i) fairly explicit descriptions of the fundamental groups of certain orbit spaces (ii) some general facts about the fundamental groups of aspherical manifolds in general and nonpositively curved manifolds in particular. This paper consists of three sections. The first provides the necessary information on orbit spaces and their fundamental groups, while the second discusses group actions on manifolds with nonzero degree maps into aspherical manifolds and the third discusses group actions on manifolds with suitable maps into higher dimensional manifolds of nonpositive curvature.