Unavoidable complexes, via an elementary equivariant index theory

Autor: Siniša T. Vrećica, Wacław Marzantowicz, Duško Jojić, Rade T. Živaljević
Rok vydání: 2020
Předmět:
Zdroj: Journal of Fixed Point Theory and Applications. 22
ISSN: 1661-7746
1661-7738
DOI: 10.1007/s11784-020-0763-2
Popis: 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.
Databáze: OpenAIRE
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