Stronger counterexamples to the topological Tverberg conjecture

08/23/2019
by   S. Avvakumov, et al.
0

Denote by Δ_N the N-dimensional simplex. A map fΔ_N→R^d is an almost r-embedding if fσ_1∩...∩ fσ_r=∅ whenever σ_1,...,σ_r are pairwise disjoint faces. A counterexample to the topological Tverberg conjecture asserts that if r is not a prime power and d>2r+1, then there is an almost r-embedding Δ_(d+1)(r-1)→R^d. We improve this by showing that if r is not a prime power and N:=(d+1)r-rd+2r+1-2, then there is an almost r-embedding Δ_N→R^d. For the r-fold van Kampen–Flores conjecture we also produce counterexamples which are stronger than previously known. Our proof is based on generalizations of the Mabillard–Wagner theorem on construction of almost r-embeddings from equivariant maps, and of the Özaydin theorem on existence of equivariant maps.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
07/18/2021

Congruence permutability is prime

We give a combinatorial proof that congruence permutability is prime in ...
research
05/25/2022

Prime Holdout Problems

This paper introduces prime holdout problems, a problem class related to...
research
08/11/2019

Linking of three triangles in 3-space

Two triples of pairwise disjoint triangles in 3-space are combinatoriall...
research
04/29/2010

Isometric Embeddings in Imaging and Vision: Facts and Fiction

We explore the practicability of Nash's Embedding Theorem in vision and ...
research
12/14/2017

Honey from the Hives: A Theoretical and Computational Exploration of Combinatorial Hives

In the first half of this manuscript, we begin with a brief review of co...
research
12/12/2019

Examples relating to Green's conjecture in low characteristics and genera

We exhibit approximately fifty Betti diagrams of free resolutions of rin...
research
08/23/2018

Substitutive structure of Jeandel-Rao aperiodic tilings

We describe the substitutive structure of Jeandel-Rao aperiodic Wang til...

Please sign up or login with your details

Forgot password? Click here to reset