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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