Using the Van Kampen obstruction
by
Eran Nevo
Cornell University

Consider the classical problem of determining whether a given simplicial complex K can be embedded in an m-dimensional Euclidian space/sphere. Van-Kampen defined a cohomological obstruction to embeddability, which was used to show that the d-skeleton of the (2d+2)-dimensional simplex (a.k.a. Van Kampen-Flores complex) does not embed in the 2d-sphere.

We use this obstruction, denoted by o(K, m), to show the following two results:

1. We define a notion of minors for simplicial complexes, generalizing the definition of minors for graphs, and show that if H is a minor of K and o(H, m) does not vanish then o(K, m) does not vanish.

2. (Joint with Uli Wagner. Generalizing Van Kampen-Flores theorem above.) Let K be the union of the d-skeleton of a 2d-dimensional piecewise linear triangulated sphere and a missing d-face of it. (Recall that a simplex F is missing in K if F is not in K and its boundary is contained in K.) Then o(K, 2d) does not vanish. Hence, K does not embed in the 2d-sphere.

We relate these two results to a conjecture due to Kalai and Sarkaria (independently) which, if true, implies the long-standing g-conjecture for simplicial spheres.

Date received: February 20, 2009