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

Positive curvature, torus symmetry, and matroids