The tropical Poincaré-Hopf theorem

Abstract

We express the beta invariant of a loopless matroid as tropical self-intersection number of the diagonal of its matroid fan (a “local” Poincaré-Hopf theorem). This provides another example of uncovering the “geometry” of matroids by expressing their invariants in terms of tropicalised geometric constructions. We also prove a global Poincaré-Hopf theorem and initiate the study of a more general tropical Lefschetz-Hopf trace formula by proving the two special cases of tropical curves and tropical tori.