Stronger Counterexamples to the Topological Tverberg Conjecture

Abstract

Denote by $$\Delta _M$$ the M-dimensional simplex. A map $$f:\Delta _M\rightarrow {{\mathbb {R}}}^d$$ is an almost r-embedding if $$f(\sigma _1)\cap \ldots \cap f(\sigma _r)=\emptyset $$ whenever $$\sigma _1,\ldots ,\sigma _r$$ are pairwise disjoint faces. A counterexample to the topological Tverberg conjecture asserts that if r is not a prime power and $$d\ge 2r+1$$ , then there is an almost r-embedding $$\Delta _{(d+1)(r-1)}\rightarrow {{\mathbb {R}}}^d$$ . This was improved by Blagojević–Frick–Ziegler using a simple construction of higher-dimensional counterexamples by taking k-fold join power of lower-dimensional ones. We improve this further (for d large compared to r): If r is not a prime power and $$N=(d+1)r-r\Big \lceil \dfrac{d+2}{r+1}\Big \rceil -2$$ , then there is an almost r-embedding $$\Delta _N\rightarrow {{\mathbb {R}}}^d$$ . The improvement follows from our stronger counterexamples to the r-fold van Kampen–Flores conjecture. 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.