On the Holonomy Group of Locally Euclidean Spaces

Abstract

Part I of this paper is devoted to giving an algebraic characterization of the fundamental group of a compact locally euclidean manifold (i.e. a compact riemann manifold with curvature and torsion tensors identically zero). If M is a compact locally euclidean manifold then the fundamental group ir of M may be considered as a subgroup of the group of rigid motions, R(n), of n-dimensional euclidean space, En, and the orbit space of xr acting on En, E /Xr, is homeomorphic to M. Bieberbach in [3] showed that if N denotes the subgroup of ir consisting of pure translations then 1. N is a free abelian group generated by n linearly independent translations. 2. FIN is a finite group. Now it is clear from the discussion in [1], that in/N is isomorphic to the holonomy group, H, of M as a riemann manifold. Also, since E'/I is a manifold, we must have 'r operating on En without fixed points. This is equivalent, assuming 1 and 2, to 3. in has no finite subgroups. To see that 7r can have no finite subgroups, we may use the P. A. Smith Theory or an elementary argument on convex sets, since we have linear transformations. The converse will be established later. In Theorem 1 we prove that a finitely generated group in is isomorphic to the fundamental group of some compact locally euclidean manifold if and only if there exists a normal subgroup N c ir which is maximal abelian (contained in no larger abelian group), where N is free on n generators, and satisfies conditions 2 and 3 above. In the second part of this paper, we will show that any finite group can be the holonomy group of a compact locally euclidean manifold. We will actually show the following: Let G be a finite group and let F/R = G, where F is a finitely generated free group and F and R are non-abelian. Then if Fo = F/[R, R] and Ro = R/[R, R], where [R, R] denotes the commutator subgroup of R, then Fo/Ro = G and Fo is isomorphic to the fundamental group of a compact locally euclidean manifold. We give an example to show that the converse is not true. The authors would like to take this opportunity to thank M. Auslander and A. Rosenberg for their suggestions and advice during the writing of this paper.