Higher minors and van Kampen's obstruction

Abstract

We generalize the notion of graph minors to all (finite) simplicial complexes. For every two simplicial complexes $H$ and $K$ and every nonnegative integer $m$, we prove that if $H$ is a minor of $K$ then the non vanishing of Van Kampen's obstruction in dimension $m$ (a characteristic class indicating non embeddability in the $(m-1)$-sphere) for $H$ implies its non vanishing for $K$. As a corollary, based on results by Van Kampen and Flores, if $K$ has the $d$-skeleton of the $(2d+2)$-simplex as a minor, then $K$ is not embeddable in the $2d$-sphere. We answer affirmatively a problem asked by Dey et. al. concerning topology-preserving edge contractions, and conclude from it the validity of the generalized lower bound inequalities for a special class of triangulated spheres.