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.
Cite
@article{arxiv.math/0602531,
title = {Higher minors and Van Kampen's obstruction},
author = {Eran Nevo},
journal= {arXiv preprint arXiv:math/0602531},
year = {2015}
}
Comments
Revised: 16 pages, 1 figure. A section on the obstruction with integer coefficients was added. In the last section, the proof of the first example (Example 6.1) was corrected, and the problem at its end was solved (Proposition 6.5). v3: Journal version, a relation between variants of the link condition, suggested by a referee, was added; see (1) and (2)