Topological representations of matroid maps

Details Topological representations of matroid maps Topological representations of matroid maps

2011-04-21

arxiv.org/abs/1104.4152v2

Mathematics

Combinatorics

24 pages, 8 figures

Scientific paper

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\"{o}m 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.

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