Fundamental groups on manifolds with positive isotropic curvature

2004/03/22 by Ailana M. Fraser
Mathematics · #math.DG

paper · pdf

published as Ann. of Math. (2), Vol. 158 (2003), no. 1, 345--354 · 10 pages published version

arxiv created 2004/03/22 · arxiv updated 2009/12/01

Abstract

A central theme in Riemannian geometry is understanding the relationships between the curvature and the topology of a Riemannian manifold. Positive isotropic curvature (PIC) is a natural and much studied curvature condition which includes manifolds with pointwisequarter-pinched sectional curvatures and manifolds with positive curvature operator. By results of Micallef and Moore there is only one topological type of compact simply connected manifold with PIC; namely any such manifold must be homeomorphic to the sphere. On the other hand, there is a large class of nonsimply connected manifolds with PIC. An important open problem has been to understand the result in this direction. We show that the fundamental group of a compact manifold Mⁿ with PIC, n geq 5, does not contain a subgroup isomorphic to ℤ ⊕ ℤ. The techniques used involve minimal surfaces.

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