On the structure of the fundamental group of manifolds with positive scalar curvature
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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 α 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 Nα of M representing α of minimum volume, compared with all other closed hypersurfaces in the homology class. Our main result is to show that the fundamental group π_1(Nα) is always virtually free. In particular, this gives rise to a new obstruction to the existence of a metric of 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 α 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 Nα of M representing α of minimum volume, compared with all other closed hypersurfaces in the homology class. Our main result is to show that the fundamental group π_1(Nα) is always virtually free. In particular, this gives rise to a new obstruction to the existence of a metric of positive scalar curvature.
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