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 $\Sigma_k$-equivariant filtration of the configuration space of $k$-points ${ \rm Conf}_k(X)$. The filtered covering map from ${ \rm Conf}_k(X,-)$ to ${ \rm Conf}_k(X,-)/\Sigma_k$ gives a canonical persistent bundle $\boldsymbol{ \xi}(X,k,-)$. We use the Stiefel-Whitney class of $\boldsymbol{ \xi}(X,k,-)$, which is in the mod $2$ persistent cohomology ring of ${ \rm Conf}_k(X,-)/\Sigma_k$, to give obstructions for $(k,r)$-regular embeddings and use the Chern class of $\boldsymbol{ \xi}(X,k,-)\otimes \mathbb{ C}$, which is in the integral persistent cohomology ring of ${ \rm Conf}_k(X,-)/\Sigma_k$, 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.
Categories: math.AT