Abstract: The classical theorem of Van Kampen and Flores states that the k-dimensional skeleton of (2k+2)-dimensional simplex cannot be embedded into R2k.
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 Rd such that for every i=0,1,...,d2−1, 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.