Algebra/Topology seminar

Speaker: Florian Frick (TU Berlin)

Title: Counterexamples to the topological Tverberg conjecture

Abstract: The "topological Tverberg conjecture" (Tverberg, 1978) 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 Bárány, Shlosman and Szücs (1981), and for prime power r by Özaydin (1987). We combine the generalized van Kampen--Shapiro--Wu theorem announced by Mabillard and Wagner (2014) with the constraint method (joint work with P. Blagojevic and G. M. Ziegler (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.