Matroid Bundles
نویسنده
چکیده
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 combinatorial analogy is remarkably strong, and has led to combinatorial results in topology and bundle-theoretic proofs in combinatorics. This paper surveys recent results on matroid bundles, and describes a canonical functor from real vector bundles to matroid bundles.
منابع مشابه
Mod 2 Cohomology of Combinatorial Grassmannians
Oriented matroids have long been of use in various areas of combinatorics [BLS93]. Gelfand and MacPherson [GM92] initiated the use of oriented matroids in manifold and bundle theory, using them to formulate a combinatorial formula for the rational Pontrjagin classes of a differentiable manifold. MacPherson [Mac93] abstracted this into a manifold theory (combinatorial differential (CD) manifolds...
متن کاملAn Ascending Vickrey Auction for Selling Bases of a Matroid
Consider selling bundles of indivisible goods to buyers with concave utilities that are additively separable in money and goods. We propose an ascending auction for the case when the seller is constrained to sell bundles whose elements form a basis of a matroid. It extends easily to polymatroids. Applications include scheduling, allocation of homogeneous goods, and spatially distributed markets...
متن کاملAn Efficient Ascending Auction
This paper proposes an ascending auction that yields an efficient outcome when the seller is restricted to sell bundles whose elements form a basis of a matroid and agents have interdependent values. This ascending auction generalizes Bikhchandani et al. (2011) who assume agents have independent private values; and Perry and Reny (2005) who study multi-unit good auctions. The key feature of the...
متن کاملHomotopy Groups of the Combinatorial Grassmannian
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 ((MacP(k; n)). The theory of oriented matroids gives rise to a combinatorial analog to the Grassmannian G(k; R n). By thinking of an oriented matroid as a \combinatorial vector space", one is led to deene t...
متن کاملParallel connections and bundles of arrangements
Let A be a complex hyperplane arrangement, and let X be a modular element of arbitrary rank in the intersection lattice of A. We show that projection along X restricts to a fiber bundle projection of the complement of A to the complement of the localization AX of A at X. The fiber is the decone of a realization of the complete principal truncation of the underlying matroid of A along the flat c...
متن کامل