Counterexamples to the topological Tverberg conjecture

By Florian Frick. 2015

71 citations recorded by the source corpus.

The “topological Tverberg conjecture” by Barany, Shlosman and Sz˝ ucs (1981) states that any continuous map of a simplex of dimension (r 1)(d + 1) to R d maps points from r disjoint faces of the simplex to the same point in R d . This was established for affine maps by Tverberg (1966), for the case when r is a prime by Barany et al., and for prime power r by Ozaydin (1987). We combine the generalized van Kampen theorem announced by Mabillard and Wagner (2014) with the constraint method of Blagojevi´ c, Ziegler and the author (2014), and thus prove the existence of counterexamples to the topological Tverberg conjecture for any number r of faces that is not a prime power. However, these counterexamples require that the dimension d of the codomain is sufficiently high: the smallest counterexample we obtain is for a map of the 100-dimensional simplex to R 19 , for r = 6.

View Paper: https://www.semanticscholar.org/paper/ffbbb4e7a9cca7834f7b8d9ed007be10ff4b902a

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1. Günter M. Ziegler
2. Pavle V. M. Blagojević
3. Fernando Muro
4. Florian Frick
5. Roman N. Karasev
6. Jirí Matousek

Citations in the corpus

• Beyond the Borsuk–Ulam Theorem: The Topological Tverberg Story (2016, methodology): “Counterexamples to the topological Tverberg conjecture Now we are going to get to a very recent piece of the topological Tverberg puzzle: We show how counterexamples to the topological Tverberg conjecture for any number of parts that his not a prime power were derived by Frick [26] [13] from the rem”