17 research outputs found
The topology of the external activity complex of a matroid
We prove that the external activity complex of a matroid
is shellable. In fact, we show that every linear extension of LasVergnas's
external/internal order on provides a shelling of
. We also show that every linear extension of LasVergnas's
internal order on provides a shelling of the independence complex
. As a corollary, and have the same -vector.
We prove that, after removing its cone points, the external activity complex is
contractible if contains as a minor, and a sphere otherwise.Comment: Comments are welcom
Finiteness theorems for matroid complexes with prescribed topology
It is known that there are finitely many simplicial complexes (up to
isomorphism) with a given number of vertices. Translating to the language of
-vectors, there are finitely many simplicial complexes of bounded dimension
with for any natural number . In this paper we study the question at
the other end of the -vector: Are there only finitely many
-dimensional simplicial complexes with for any given ? The
answer is no if we consider general complexes, but when focus on three cases
coming from matroids: (i) independence complexes, (ii) broken circuit
complexes, and (iii) order complexes of geometric lattices. We prove the answer
is yes in cases (i) and (iii) and conjecture it is also true in case (ii).Comment: to appear in European Journal of Combinatoric
The topology of the external activity complex of a matroid
International audienceWe prove that the external activity complex Act<(M) of a matroid is shellable. In fact, we show that every linear extension of Las Vergnas's external/internal order <ext/int on M provides a shelling of Act<(M). We also show that every linear extension of Las Vergnas's internal order <int on M provides a shelling of the independence complex IN(M). As a corollary, Act<(M) and M have the same h-vector. We prove that, after removing its cone points, the external activity complex is contractible if M contains U3,1 as a minor, and a sphere otherwise
Relaxations of the matroid axioms I: Independence, Exchange and Circuits
International audienceMotivated by a question of Duval and Reiner about higher Laplacians of simplicial complexes, we describe various relaxations of the defining axioms of matroid theory to obtain larger classes of simplicial complexes that contain pure shifted simplicial complexes. The resulting classes retain some of the matroid properties and allow us to classify matroid properties according to the relevant axioms needed to prove them. We illustrate this by discussing Tutte polynomials. Furthermore, we extend a conjecture of Stanley on h-vectors and provide evidence to show that the extension is better suited than matroids to study the conjecture