We extend the notion of a minor from matroids to simplicial complexes. We show that the class of matroids, as well as the class of independence complexes, is characterized by a single forbidden minor. Inspired by a recent result of Aharoni and Berger, we investigate possible ways to extend the matroid intersection theorem to simplicial complexes.

We consider topological aspects of decision trees on simplicial complexes, concentrating on how to use decision trees as a tool in topological combinatorics. By Robin Forman’s discrete Morse theory, the number of evasive faces of a given dimension i with respect to a decision tree on a simplicial complex is greater than or equal to the ith reduced Betti number (over any field) of the complex. U...

In this paper, we review a method for computing and parameterizing the set of homotopy classes of chain maps between two chain complexes. This is then applied to finding topologically meaningful maps between simplicial complexes, which in the context of topological data analysis, can be viewed as an extension of conventional unsupervised learning methods to simplicial complexes.

We introduce a notion of join for (augmented) simplicial sets gen-eralising the classical join of geometric simplicial complexes. The definition comes naturally from the ordinal sum on the base simplicial category ∆.

A notion of an i-banner simplicial complex is introduced. For various values of i, these complexes interpolate between the class of flag complexes and the class of all simplicial complexes. Examples of simplicial spheres of an arbitrary dimension that are (i+1)-banner but not i-banner are constructed. It is shown that several theorems for flag complexes have appropriate i-banner analogues. Amon...

It is shown that conditions stronger in a certain sense than colorshifting cannot be placed on the class of colored complexes without changing the characterization of the flag f-vectors. In the late 1970s, Stanley [6] showed that two particular classes of simplicial complexes have equivalent characterizations of their flag f-vectors. Several years later, Björner, Frankl, and Stanley [1] showed ...