Beyond the Borsuk–Ulam Theorem: The Topological Tverberg Story

By Pavle V. M. Blagojević, Günter M. Ziegler. 2016

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Barany’s “topological Tverberg conjecture” from 1976 states that any continuous map of an N-simplex \(\Delta _{N}\) to \(\mathbb{R}^{d}\), for N ≥ (d + 1)(r − 1), maps points from r disjoint faces in \(\Delta _{N}\) to the same point in \(\mathbb{R}^{d}\). The proof of this result for the case when r is a prime, as well as some colored version of the same result, using the results of Borsuk–Ulam and Dold on the non-existence of equivariant maps between spaces with a free group action, were main topics of Matousek’s 2003 book “Using the Borsuk–Ulam theorem.”

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ff300c27295cbe9380e5749eb77779932dd1006c