Abstract
Denote by Δ M the M-dimensional simplex. A map f: Δ M→ Rd is an almost r-embedding if f(σ1) ∩ … ∩ f(σr) = ∅ whenever σ1, … , σr are pairwise disjoint faces. A counterexample to the topological Tverberg conjecture asserts that if r is not a prime power and d≥ 2 r+ 1 , then there is an almost r-embedding Δ (d+1)(r-1)→ Rd . 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⌈d+2r+1⌉-2 , then there is an almost r-embedding Δ N→ Rd . 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.
Original language | English |
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Pages (from-to) | 717-727 |
Number of pages | 11 |
Journal | Combinatorica |
Volume | 43 |
Issue number | 4 |
DOIs | |
State | Published - Aug 2023 |
Externally published | Yes |
Keywords
- Deleted product obstruction
- Equivariant maps
- Multiple points of maps
- The topological Tverberg conjecture
All Science Journal Classification (ASJC) codes
- Discrete Mathematics and Combinatorics
- Computational Mathematics