Year: 2025
Type: preprint
Abstract: 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 ${\rm Conf}_k(X,r)$, $r\geq 0$, give a $Σ_k$-equivariant filtration of the configuration space of $k$-points ${\rm Conf}_k(X)$. The filtered covering map from ${\rm Conf}_k(X,-)$ to ${\rm Conf}_k(X,-)/Σ_k$ 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 ${\rm Conf}_k(X,-)/Σ_k$, to give obstructions for $(k,r)$-regular embeddings and use the Chern class of $\boldsymbolξ(X,k,-)\otimes \mathbb{C}$, which is in the integral persistent cohomology ring of ${\rm Conf}_k(X,-)/Σ_k$, 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.
Source: arXiv (Cornell University)
Author: Ren, Shiquan
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4407424848