Fundamental groups of manifolds with positive isotropic curvature

By A. Fraser. 2003

48 citations recorded by the source corpus.

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 pointwise quarter-pinched sectional curvatures and manifolds with positive curvature operator. By the 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 fundamental groups of manifolds with PIC. In this paper we prove a new result in this direction. We show that the fundamental group of a compact manifold M n with PIC, n ≥ 5, does not contain a subgroup isomorphic to ⊕ . The techniques used involve minimal surfaces.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e784bfde99424b6f139f3c75ab7351097b4d2863

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Manifolds with 1/4-pinched curvature are space forms (2007, background): “Fraser [7] on nonsimply connected manifolds with positive isotropic curvature suggest that Hamilton’s four-dimensional flow with surgeries may hold in all dimensions.”
Complete classification of compact four-manifolds with positive isotropic curvature (2008, background): “Another interesting result for higher dimensional Riemannian manifold with positive isotropic curvature is the result of Fraser and Wolfson [7] [8] who proved that the fundamental group of any compact surface of genus g ≥ 1 cannot occur as a subgroup of such manifold when its dimension is greater th”
Ricci flow with surgery on manifolds with positive isotropic curvature (2017, background): “Earlier results on the topology of manifolds with positive isotropic curvature (which are based on minimal surface techniques) are discussed in [14],[15],[28].”