Topological representations of matroid maps

TL;DR

This paper extends the topological representation of matroids to include structure-preserving maps, showing that weak maps induce continuous topological maps and establishing a functorial relationship.

Contribution

It demonstrates that weak maps between matroids induce continuous maps between their topological representations, generalizing known results from oriented matroids.

Findings

Weak maps induce continuous topological maps

The construction is functorial from matroids to topological spaces

Provides a new proof of a Whitney number result

Abstract

The Topological Representation Theorem for (oriented) matroids states that every (oriented) matroid can be realized as the intersection lattice of an arrangement of codimension one homotopy spheres on a homotopy sphere. In this paper, we use a construction of Engstr\"om to show that structure-preserving maps between matroids induce topological mappings between their representations; a result previously known only in the oriented case. Specifically, we show that weak maps induce continuous maps and that the process is a functor from the category of matroids with weak maps to the homotopy category of topological spaces. We also give a new and conceptual proof of a result regarding the Whitney numbers of the first kind of a matroid.