Barycenters of Polytope Skeleta and Counterexamples to the Topological Tverberg Conjecture, via Constraints

TL;DR

This paper employs the constraints method to provide a concise proof of a representation result for points in polytopes and demonstrates that the Topological Tverberg Conjecture fails for non-prime power r, revealing significant counterexamples.

Contribution

It introduces a new proof technique for polytope barycenter representations and shows the failure of the Topological Tverberg Conjecture for non-prime powers using the Whitney trick.

Findings

Short proof of Dobbins' 2015 result on polytope barycenters

Counterexamples to the Topological Tverberg Conjecture for non-prime powers

Implication of the Whitney trick in topological intersection problems

Abstract

Using the authors' 2014 "constraints method," we give a short proof for a 2015 result of Dobbins on representations of a point in a polytope as the barycenter of points in a skeleton, and show that the "r-fold Whitney trick" of Mabillard and Wagner (2014/2015) implies that the Topological Tverberg Conjecture for r-fold intersections fails dramatically for all r that are not prime powers.