Positive curvature, torus symmetry, and matroids

2022/12/15 by Kennard, Lee, Wiemeler, Michael, Wilking, Burkhard · 3 citations
#05B35 (Secondary) #53C20 (Primary) 05C35 #Combinatorics (math.CO) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2212.08152

Abstract

We identify a link between regular matroids and torus representations all of whose isotropy groups have an odd number of components. Applying Seymour's 1980 classification of the former objects, we obtain a classification of the latter. In addition, we prove optimal upper bounds for the cogirth of regular matroids up to rank nine, and we apply this to prove the existence of fixed-point sets of circles with large dimension in a torus representation with this property up to rank nine. Finally, we apply these results to prove new obstructions to the existence of Riemannian metrics with positive sectional curvature and torus symmetry.

Cited by

• ℤ_p-torus actions on positively curved manifolds
• On Hopf's conjecture and positive second intermediate Ricci curvature
• Cubic torus obstructions of small Betti number

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