Understand
B\'ar\'any's "topological Tverberg conjecture" from 1976 states that any continuous map of an $N$-simplex $\Delta_N$ to $\mathbb{R}^d$, for $N\ge(d+1)(r-1)$, 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 Matou\v{s}ek's 2003 book "Using the Borsuk-Ulam theorem." In this paper we show how advanced equivariant topology methods allow one to go beyond the prime case of the topological Tverberg conjecture.
- First we explain in detail how equivariant cohomology tools (employing the Borel construction, comparison of Serre spectral sequences, Fadell-Husseini index, etc.) can be used to prove the topological Tverberg conjecture whenever $r$ is a prime power.
- Our presentation includes a number of improved proofs as well as new results, such as a complete determination of the Fadell-Husseini index of chessboard complexes in the prime case.