Topological Invariants and Curvature

Abstract. It is widely known that the fundamental group of a Lie group, and in general a symmetric space, is abelian. In the current paper it is demonstrated that any finitely generated abelian group is the fundamental group of a compact Lie group. In addition, it is proved that for any arbitrary group there is a differentiable manifold of dimension greater than 3 whose fundamental group is that arbitrary group.