Persistent bundles over configuration spaces and obstructions for regular embeddings

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 ${\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{f}}_k(X,r)$, $r\ge 0$, give a ${\mathrm{\Sigma }}_k$-equivariant filtration of the configuration space of $k$-points ${\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{f}}_k(X)$. The filtered covering map from ${\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{f}}_k(X,-)$ to ${\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{f}}_k(X,-)\mathrm{/}{\mathrm{\Sigma }}_k$ gives a canonical persistent bundle $\mathit{\xi }(X,k,-)$. We use the Stiefel-Whitney class of $\mathit{\xi }(X,k,-)$, which is in the mod $2$ persistent cohomology ring of ${\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{f}}_k(X,-)\mathrm{/}{\mathrm{\Sigma }}_k$, to give obstructions for $(k,r)$-regular embeddings and use the Chern class of $\mathit{\xi }(X,k,-)\otimes \mathbb{C}$, which is in the integral persistent cohomology ring of ${\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{f}}_k(X,-)\mathrm{/}{\mathrm{\Sigma }}_k$, 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.