Hebrew University topology and geometry seminar

Abstract: The classical theorem of Van Kampen and Flores states that the k-dimensional skeleton of (2k+2)-dimensional simplex cannot be embedded into $\mathbb{R}^{2k}$.
We present a version of this theorem for chain maps and as an application we prove a qualitative topological Helly-type theorem.

If we define the Helly number of a finite family of sets to be one if all sets in the family have a point in common and as the largest size of inclusion-minimal subfamily with empty intersection otherwise, the theorem can be stated as follows:
There exists a function $h(b,d)$ such that given a finite family $F$ of sets in $\mathbb{R}^d$ such that for every $i=0,1,..., \frac{d}{2}-1$, and every subfamily $G$ of $F$ the intersection of $G$ has i-th Betti number at most b, then the Helly number of $F$ is bounded by above by $h(b,d)$.

This work is a part of a systematic effort to replace various homotopic assumptions with more tractable homological ones.