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

Title & Authors
ON THE STRUCTURE OF THE FUNDAMENTAL GROUP OF MANIFOLDS WITH POSITIVE SCALAR CURVATURE
Kim, Jin-Hong; Park, Han-Chul;

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 $\small{\leq}$ n $\small{\leq}$ 7) with positive scalar curvature and non-trivial first Betti number, and let be $\small{\alpha}$ non-trivial codimension one homology class in $\small{H_{n-1}}$(M;$\small{\mathbb{R}}$). Then it is known as in [8] that there exists a closed embedded hypersurface $\small{N_{\alpha}}$ of M representing $\small{\alpha}$ of minimum volume, compared with all other closed hypersurfaces in the homology class. Our main result is to show that the fundamental group $\small{{\pi}_1(N_{\alpha})}$ 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;
Language
English
Cited by
1.
ADDENDUM AND ERRATUM TO "ON THE STRUCTURE OF THE FUNDAMENTAL GROUP OF MANIFOLDS WITH POSITIVE SCALAR CURVATURE",;

대한수학회보, 2013. vol.50. 2, pp.537-542
1.
ADDENDUM AND ERRATUM TO "ON THE STRUCTURE OF THE FUNDAMENTAL GROUP OF MANIFOLDS WITH POSITIVE SCALAR CURVATURE", Bulletin of the Korean Mathematical Society, 2013, 50, 2, 537
References
1.
M. Dunwoody, The accessibility of finitely presented groups, Invent. Math. 81 (1985), no. 3, 449-457.

2.
A. Fraser, Fundamental groups of manifolds with positive isotropic curvature, Ann. of Math. (2) 158 (2003), no. 1, 345-354.

3.
A. Fraser and J. Wolfson, The fundamental group of manifolds of positive isotropic curvature and surface groups, Duke Math. J. 133 (2006), no. 2, 325-334.

4.
M. Gromov, Positive curvature, macroscopic dimension, spectral gaps and higher signatures, Functional analysis on the eve of the 21st century, Vol. II (New Brunswick, NJ, 1993), 1-213, Progr. Math., 132, Birkhauser Boston, Boston, MA, 1996.

5.
M. Gromov and H. Lawson, Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Inst. Hautes Etudes Sci. Publ. Math. No. 58 (1983), 83-196.

6.
R. Hamilton, Four-manifolds with positive isotropic curvature, Comm. Anal. Geom. 5 (1997), no. 1, 1-92.

7.
J. Kazdan and F. Warner, Prescribing curvatures, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973), Part 2, pp. 309-319. Amer. Math. Soc., Providence, R.I., 1975.

8.
H. B. Lawson Jr., Minimal varieties in real and complex geometry, Seminaire de Mathematiques Superieures, No. 57 (Ete 1973). Les Presses de l'Universite de Montreal, Montreal, Que., 1974. 100 pp.

9.
M. Micallef and J. Moore, Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes, Ann. of Math. (2) 127 (1988), no. 1, 199-227.

10.
M. Micallef and M. Wang, Metrics with nonnegative isotropic curvature, Duke Math. J. 72 (1993), no. 3, 649-672.

11.
M. Ramachandran and J. Wolfson, Fill radius and the fundamental group, Journal of Topology and Analysis 2 (2010), 99-107.

12.
R. Schoen and S. T. Yau, On the structure of manifolds with positive scalar curvature, Manuscripta Math. 28 (1979), no. 1-3, 159-183.

13.
R. Schoen and S. T. Yau, The existence of a black hole due to condensation of matter, Comm. Math. Phys. 90 (1983), no. 4, 575-579.

14.
J.-P. Serre, Trees, Springer-Verlag, Berlin, 1980.

15.
J. Wolfson, Four manifolds with two-positive Ricci curvature, preprint (2008), arXiv: 0805.4183v2.