ON THE STRUCTURE OF THE FUNDAMENTAL GROUP OF MANIFOLDS WITH POSITIVE SCALAR CURVATURE

Abstract

The aim of this paper is to study the structure of the fundamental group of a closed oriented Riemannian manifold with positive scalar curvature. To be more precise, let M be a closed oriented Riemannian manifold of dimension n (4 <= n <= 7) with positive scalar curvature and non-trivial first Betti number, and let a be a non-trivial codimension one homology class in H(n-1)(M; R). Then it is known as in [8] that there exists a closed embedded hypersurface N(alpha) of M representing alpha of minimum volume, compared with all other closed hypersurfaces in the homology class. Our main result is to show that the fundamental group pi(1)(N(alpha)) is always virtually free. In particular, this gives rise to a new obstruction to the existence of a metric of positive scalar curvature.