A homotopy-theoretic approach to the topological Tverberg conjecture

Let \(r \geq 2\) and \(d \geq 1\) be integers, let \(N=(d+1)(r-1)\), and let \(\Delta^N\) denote a standard \(N\)-simplex. The Topological Tverberg Conjecture states that for any continuous map \(f: \Delta^N \rightarrow \mathbb{R}^d\), there are \(r\) pairwise disjoint faces \(\sigma_1,\ldots ,\sigma_r\) of \(\Delta^N\) such that \(f(\sigma_1) \cap \ldots \cap f(\sigma_r)\) is non-empty. F. Frick announced a counterexample to the conjecture for \(d\geq 3r+1\), when \(r\) is not a power of a prime, and other counterexamples have also been found. My talk will discuss an alternative analysis of these counterexamples using the manifold calculus of functors, a technique that we hope will provide insight into the minimal counterexample and other related questions. (Joint work with Ismar Volic and Franjo Sarcevic.)