Unavoidable complexes, via an elementary equivariant index theory

Abstract

The partition invariant π(K) of a simplicial complex K⊆ 2 [m] is the minimum integer ν, such that for each partition A1⊎ ⋯ ⊎ Aν= [m] of [m], at least one of the sets Ai is in K. A complex K is r-unavoidable if π(K) ≤ r. We say that a complex K is almost r-non-embeddable in Rd if, for each continuous map f: | K| → Rd, there exist r vertex disjoint faces σ1, ⋯ , σr of | K| , such that f(σ1) ∩ ⋯ ∩ f(σr) ≠ ∅. One of our central observations (Theorem 2.1), summarizing and extending results of Schild et al. is that interesting examples of (almost) r-non-embeddable complexes can be found among the joins K= K1∗ ⋯ ∗ Ks of r-unavoidable complexes.