Higher minors and Van Kampen's obstruction

Details Higher minors and Van Kampen's obstruction Higher minors and Van Kampen's obstruction

2006-02-23

arxiv.org/abs/math/0602531v2

Math. Scand. 101 (2007), no. 2, 161--176.

Mathematics

Combinatorics

Revised: 16 pages, 1 figure. A section on the obstruction with integer coeficients was added. In the last section, the proof o

Scientific paper

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.

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