2016/03/27 by Rade T. Živaljević, Živaljević, Rade T.
Mathematics · #06B35 #18B35 #52-02 #52C35 #55R65 #55R80 #Advanced Combinatorial Mathematics #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT)
paper · pdf · doi:10.48550/arxiv.1603.08175
openalex publication_date 2016/03/27 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
This is both an expository and research paper where we advocate a systematic\nstudy of continuous analogues of finite partially ordered sets, convex\npolytopes, oriented matroids, arrangements of subspaces, finite simplicial\ncomplexes, and other combinatorial structures. Among the illustrative examples\nare an Euler formula for a class of `continuous convex polytopes' (conjectured\nby Kalai and Wigderson), a duality result for a class of `continuous matroids',\na calculation of the Euler characteristic of ideals in the Grassmannian poset\n(related to a problem of Gian-Carlo Rota), an exposition of the `homotopy\ncomplementation formula' for topological posets and its relation to the results\nof Kallel and Karoui about `weighted barycenter spaces' and a conjecture of\nVassiliev about simplicial resolutions of singularities. We also include an\nextension of the index inequality (Sarkaria's inequality) based on interpreting\ndiagrams of spaces as continuous posets.\n