  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.
