Stronger counterexamples to the topological Tverberg conjecture

Abstract

Denote by ∆. . , 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 ∆ (d+1)(r−1) → R d . We improve this by showing that if r is not a prime power and N := (d + 1)r − r d + 2 r + 1 − 2, then there is an almost r-embeddingFor 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.

Envy‐free Division Using Mapping Degree

“…Some remarks and consequences. We first answer some doubts expressed by an unknown referee of [2] about the novelty of Theorem 5.1. Our Theorem 5.1 is similar to, but is not a particular case of [5,Theorem 3.6].…”

“…In terms of the works [8,12], the theorems of this section show that the direct approach to fair partition problems does not only fail in terms of the primary cohomology obstruction but also in terms of higher obstructions, when n is odd and not a prime power. This approach (and its appropriate generalizations) has some particular consequences for the topological Tverberg problem (or, more generally, to van Kampen-Flores-type problems), which are given in [2].…”

“…This result is not related to the original segment envy-free division problem, but it prevents using some of the well-known general techniques for other envy-free division or fair partition problems. Its analogues and their consequences are developed in [2,3]. §2.…”

The topological Tverberg problem beyond prime powers

“…Avvakumov, Karasev, and Skopenkov [AKS19] show that if q is not a prime power then there exists a continuous map f : ∆ n → R d with f (σ 1 ) ∩ • • • ∩ f (σ q ) = ∅ for any q pairwise disjoint faces σ 1 , . .…”