Almost-embeddability into manifolds and Helly-type theorems

A continuous map f : K → X of a simplicial complex into a topological space X is an almost-embedding if any two disjoint faces σ1, σ2 of K satisfy f(σ1) ∩ f(σ2) = ∅. Almost-embeddings generalize embeddings of simplicial complexes in a direction which is more suitable for combinatorial applications, for example for proving Helly type theorems.

Helly-type theorems are statements of the type “If a finite family F of sets satisfies property P, and the intersection of every h sets in F is non-empty, then the intersection of all sets in F is non-empty”. The most typical example of such theorem is the original Helly’s theorem, where P is the statement “all members of the family are convex sets in R^d” and h = d + 1.

For certain simplicial complexes, namely complete skeleta of (large enough) simplices, the non-existence of the almost embedding into a topo- logical space X guarantees existence of tight Helly type theorems for X.

In this talk we give some bounds for almost embeddings of k-dimensional skeleta into 2k-manifolds. As a consequence we obtain several tight Helly type theorems.