Topological properties of activity orders for matroid bases

Las Vergnas (European J. Combin. 22 (2001) 709) introduced several lattice structures on the bases of an ordered matroid M by using their external and internal activities. He also noted (personal communication) that when computing the Möbius function of these lattices, it was often zero, although he had no explanation for that fact. The purpose of this paper is to provide a topological reason for this phenomenon. In particular, we show that the order complex of the external lattice L(M) is homotopic to the independence complex of the restriction M*|T where M* is the dual of M and T is the top element of L(M). We then compute some examples showing that this latter complex is often contractible which forces all its homology groups, and thus its Möbius function, to vanish. A theorem of Björner (Matroid Applications, Encyclopedia of Mathematics and its Applications, vol. 40, Cambridge University Press, Cambridge, 1992, pp. 226.) also helps us to calculate the homology of the matroid complex.