  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.
