We express the beta invariant of a loopless matroid as tropical
self-intersection number of the diagonal of its matroid fan (a "local"
Poincar\'e-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 Poincar\'e-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.Comment: 27 pages, 3 figure