2025/02/11 by Shiquan Ren, Ren, Shiquan · 1 citation
Mathematics · Medicine · #Algebraic Topology (math.AT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Intracranial Aneurysms: Treatment and Complications
paper · pdf · doi:10.48550/arxiv.2502.07476
openalex publication_date 2025/02/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We construct persistent bundles over configuration spaces of hard spheres and use the characteristic classes of these persistent bundles to give obstructions for embedding problems. The configuration spaces of k-hard spheres \rm Confk(X,r), r≥ 0, give a Σk-equivariant filtration of the configuration space of k-points \rm Confk(X). The filtered covering map from \rm Confk(X,-) to \rm Confk(X,-)/Σk gives a canonical persistent bundle \boldsymbolξ(X,k,-). We use the Stiefel-Whitney class of \boldsymbolξ(X,k,-), which is in the mod 2 persistent cohomology ring of \rm Confk(X,-)/Σk, to give obstructions for (k,r)-regular embeddings and use the Chern class of \boldsymbolξ(X,k,-)⊗ ℂ, which is in the integral persistent cohomology ring of \rm Confk(X,-)/Σk, to give obstructions for complex (k,r)-regular embeddings. As applications, we discuss the geometric realizations of the independence complexes given by the regular embeddings. With the help of the persistent homology tools, the k-regular embedding problems of manifolds, the sphere-packing problems on manifolds, and the geometric realization problems of the independence complexes of graphs are prospectively to be computed approximately.