On the structure of the fundamental group of manifolds with positive scalar curvature
Bull. Korean Math. Soc. 2011 Vol. 48, No. 1, 129-140
https://doi.org/10.4134/BKMS.2011.48.1.129
Published online January 1, 2011
Jin Hong Kim and Han Chul Park
Korea Advanced Institute of Science and Technology, Korea Advanced Institute of Science and Technology
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 $n$ ($4\le n \le 7$) with positive scalar curvature and non-trivial first Betti number, and let $\alpha$ be a non-trivial codimension one homology class in $H_{n-1}(M; {\mathbb R})$. 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.
Keywords : fundamental group, positive scalar curvature
MSC numbers : 57C53
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