Higher Minors and Van Kampen’s Obstruction

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 [19] and Flores [5], 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. [3] concerning topology-preserving edge contractions, and conclude from it the validity of the generalized lower bound inequalities for a special class of triangulated spheres.