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 [20] 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.

Algebraic shifting of strongly edge decomposable spheres

“…The above result was proved in [24,Corollary 4.3]. Actually, the h-vector of strongly edge decomposable complexes satisfies a stronger condition.…”

“…Then the link of Γ with respect to any face F ∈ Γ is again a PL-sphere. Also, it was proved in [24,Theorem 1.4] that if Γ satisfies the Link condition with respect to {i, j} ∈ Γ then C Γ (ij) is also a PL-sphere. These facts and Proposition 3.2 may help to study the strong Lefschetz property of PL-spheres.…”

The Lefschetz property for barycentric subdivisions of shellable complexes

“…This can be seen by using the inverse stellar moves starting with the last simplicial complex in the sequence of complexes and moving backwards. The notion of strongly edge decomposable complexes was introduced in [10].…”

Spheres arising from multicomplexes

“…Edge decomposability was introduced by Nevo [21] in the study of g-vectors of simplicial spheres. This property is important since if a simplicial complex is edge decomposable then its face vector satisfies McMullen's g-condition.…”