Beyond the Borsuk–Ulam theorem

by · Jul 23, 2016 · 343 views ·

Beyond the Borsuk–Ulam theorem: The topological Tverberg story Bárány's "topological Tverberg conjecture'' from 1976 states that any continuous map of an N-simplex ΔN to Rd, for N≥(d+1)(r−1), maps r points from disjoint faces in ΔN to the same point in Rd. 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 equivariant maps between spaces with a free group action, were main topics of Matoušek's 2003 book "Using the Borsuk–Ulam theorem.'' In this lecture we go beyond, and present the methods that yield the case when r is a prime power (and give the proof!), exhibit the failure of configuration test map scheme for non-prime powers, introduce the "constraint method'' which yields a great variety of variations and corollaries, and sketch the path towards the Optimal colored Tverberg theorem and Bárány-Larman conjecture. (This lecture is motivated by the classical book of Jiří Matoušek "Using the Borsuk–Ulam Theorem. Lectures on Topological Methods in Combinatorics and Geometry'', and is based on the joint articles with Imre Bárány and Günter Ziegler.)

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