A knot characterization and 1–connected nonnegatively curved 4–manifolds with circle symmetry

2013/04/17 by Karsten Grove, Burkhard Wilking · 32 citations
Mathematics · #Curvature #Equivariant map #Extension (predicate logic) #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Knot (papermaking) #Orbit (dynamics) #Symmetry (geometry) #math.DG #msc:53C21

paper · pdf · doi:10.2140/gt.2014.18.3091

published in Geometry & Topology 18(5), 3091-3110 (Mathematical Sciences Publishers) · 13 p

arxiv created 2013/04/17 · openalex publication_date 2014/12/01 · arxiv updated 2016/01/20 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

We classify nonnegatively curved simply connected 4-manifolds with circle symmetry up to equivariant diffeomorphisms. The main problem is to rule out knotted curves in the singular set of the orbit space. As an extension of this work we classify all knots in S 3 that can be realized as an extremal set with respect to an inner metric on S 3 that has nonnegative curvature in the Alexandrov sense.

Citations

• The topology of four-dimensional manifolds
• Three-manifolds with positive Ricci curvature
• On the curvature of biquotients
• Perelman’s Stability Theorem