Positive curvature, torus symmetry, and matroids

TL;DR

This paper links regular matroids with torus representations having odd component isotropy groups, classifies these objects, and applies the results to geometric problems involving positive curvature and symmetry.

Contribution

It provides a classification of certain torus representations via regular matroids and establishes new bounds and obstructions related to positive curvature and symmetry.

Findings

Classified torus representations with odd isotropy group components using matroid theory.

Established optimal upper bounds for the cogirth of regular matroids up to rank nine.

Proved new obstructions to positive sectional curvature metrics with torus symmetry.

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.