Authors: Jojić, Duško
Marzantowicz, Wacław
Vrećica, Siniša
Živaljević, Rade 
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Unavoidable complexes, via an elementary equivariant index theory
Journal: Journal of Fixed Point Theory and Applications
Volume: 22
Issue: 2
Issue Date: 1-Jun-2020
Rank: M21a
ISSN: 1661-7738
DOI: 10.1007/s11784-020-0763-2
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.
Keywords: equivariant index theory | Partition invariant | unavoidable complexes
Publisher: Springer Link
Project: Geometry and Topology of Manifolds, Classical Mechanics and Integrable Dynamical Systems 
Topology, geometry and global analysis on manifolds and discrete structures 

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