In the absence of a proof of this conjecture, much hard work has been done to prove weaker results in the combinatorial theory of subdivision. Alexander ([Ale30]) proved that if the underlying space… (More)

Matroid bundles, introduced by MacPherson, are combinatorial analogues of real vector bundles. This paper sets up the foundations of matroid bundles. It defines a natural transformation from… (More)

We prove that the homotopy groups of the oriented matroid Grass-mannian MacP(k; n) are stable as n ! 1, that 1 ((MacP(k; n)) = 1 (G(k; R n)), and that there is a surjection 2 (G(k; R n)) ! 2… (More)

The Grassmannian G!k!!n# of k-planes in !n is a smooth manifold, hence can be triangulated. Identify !n as a subspace of !n+1, and let !∞ be the union (colimit) of the !n’s. The Grassmannian G!k!!∞#… (More)

For any rank r oriented matroid M , a construction is given of a ”topological representation” of M by an arrangement of homotopy spheres in a simplicial complex which is homotopy equivalent to Sr−1.… (More)

Combinatorial vector bundles, or matroid bundles, are a combinatorial analog to real vector bundles. Combinatorial objects called oriented matroids play the role of real vector spaces. This… (More)

We prove that all combinatorial diierential manifolds involving only Euclidean oriented matroids are PL manifolds. In doing so we introduce a new notion of triangulations of oriented matroids, and… (More)