Tverberg's theorem with constraints
Autor: | Stephan Hell |
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Rok vydání: | 2007 |
Předmět: |
05A18
52A37 Continuous map Disjoint sets Computer Science::Computational Geometry Mathematics::Algebraic Topology Equivariant method Theoretical Computer Science Combinatorics Intersection FOS: Mathematics Discrete Mathematics and Combinatorics Mathematics - Combinatorics Mathematics::Metric Geometry Prime power Mathematics Discrete mathematics Conjecture Mathematics::Combinatorics Tverberg's theorem Sierkma's conjecture Topological Tverberg theorem Radon's theorem Computational Theory and Mathematics Affine transformation Combinatorics (math.CO) |
DOI: | 10.48550/arxiv.0704.2713 |
Popis: | The topological Tverberg theorem claims that for any continuous map of the (q-1)(d+1)-simplex to R^d there are q disjoint faces such that their images have a non-empty intersection. This has been proved for affine maps, and if $q$ is a prime power, but not in general. We extend the topological Tverberg theorem in the following way: Pairs of vertices are forced to end up in different faces. This leads to the concept of constraint graphs. In Tverberg's theorem with constraints, we come up with a list of constraints graphs for the topological Tverberg theorem. The proof is based on connectivity results of chessboard-type complexes. Moreover, Tverberg's theorem with constraints implies new lower bounds for the number of Tverberg partitions. As a consequence, we prove Sierksma's conjecture for $d=2$, and $q=3$. Comment: 16 pages, 12 figures. Accepted for publication in JCTA. Substantial revision due to the referees |
Databáze: | OpenAIRE |
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