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The fundamental group of manifolds of positive isotropic curvature and surface groups. (English) Zbl 1110.53027
The authors study the topology of compact manifolds with positive isotropic curvature. There are many examples of nonsimply connected compact manifolds with positive isotropic curvature. By using stable minimal surface theory, the authors prove that the fundamental group of a compact Riemannian manifold of dimension at least 5 with positive isotropic curvature does not contain a subgroup isomorphic to the fundamental group of a compact Riemann surface. The main theorem of the paper is as follows: Let \(M\) be a compact \(n\)-manifold, \(n\geq 5\), with positive isotropic curvature. Then the fundamental group \(\pi_1 (M)\) of \(M\) does not contain a subgroup isomorphic to the fundamental group \(\pi_1(\Sigma_0)\) of a compact surface \(\Sigma_0\) of genus \(g\geq 1\). Moreover, the authors propose the following conjecture: The fundamental group of a compact Riemannian manifold with positive isotropic curvature has, virtually, no subgroup with exactly one end.

MSC:
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58E12 Variational problems concerning minimal surfaces (problems in two independent variables)
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References:
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