Bulletin of the Korean Mathematical Society (대한수학회보)

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

  • Kim, Jin-Hong (DEPARTMENT OF MATHEMATICAL SCIENCES KOREA ADVANCED INSTITUTE OF SCIENCE AND TECHNOLOGY) ;
  • Park, Han-Chul (DEPARTMENT OF MATHEMATICAL SCIENCES KOREA ADVANCED INSTITUTE OF SCIENCE AND TECHNOLOGY)
  • Received : 2009.05.21
  • Published : 2011.01.31
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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 $\leq$ n $\leq$ 7) with positive scalar curvature and non-trivial first Betti number, and let be $\alpha$ 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.

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References

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Cited by

  1. ADDENDUM AND ERRATUM TO "ON THE STRUCTURE OF THE FUNDAMENTAL GROUP OF MANIFOLDS WITH POSITIVE SCALAR CURVATURE" vol.50, pp.2, 2013, https://doi.org/10.4134/BKMS.2013.50.2.537