292 research outputs found
Topological representations of matroid maps
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.Comment: Final version, 21 pages, 8 figures; Journal of Algebraic
Combinatorics, 201
Positively oriented matroids are realizable
We prove da Silva's 1987 conjecture that any positively oriented matroid is a
positroid; that is, it can be realized by a set of vectors in a real vector
space. It follows from this result and a result of the third author that the
positive matroid Grassmannian (or positive MacPhersonian) is homeomorphic to a
closed ball.Comment: 20 pages, 3 figures, references adde
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