Unavoidable complexes, via an elementary equivariant index theory

Abstract

The partition invariant $\pi (K)$ of a simplicial complex $K\subseteq 2^{[m]}$ is the minimum integer $\nu$, such that for each partition $A_1\uplus \cdots \uplus A_\nu = [m]$ of [m], at least one of the sets $A_i$ is in K. A complex K is r-unavoidable if $\pi (K)\le r$. We say that a complex K is almost r-non-embeddable in ${\mathbb {R}}^d$ if, for each continuous map $f: \vert K\vert \rightarrow {\mathbb {R}}^d$, there exist r vertex disjoint faces $\sigma _1,\cdots , \sigma _r$ of $\vert K\vert$, such that $f(\sigma _1)\cap \cdots \cap f(\sigma _r)\ne \emptyset$. 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 $K = K_1*\cdots *K_s$ of r-unavoidable complexes.