Attention, Topological Combinatorialists! The topological Tverberg Conjecture, described as ‘a holy grail of topological combinatorics’, is false.

The conjecture says that any continuous map of a simplex of dimension $(r−1)(d+1)$ to $\mathbb{R}^d$ maps points from $r$ disjoint faces of the simplex to the same point in $\mathbb{R}^d$. In certain cases the conjecture has been proven true, but there have been found counterexamples in the case where $r$ is not a prime power, for sufficiently large values of $d$: the smallest counterexample found is for a map of the 100-dimensional simplex to $\mathbb{R}^{19}$, with $r=6$.

The result was recently unveiled at the Oberwolfach Maths Research Institute, which is situated in the Black Forest in Germany and regularly hosts bands of fiercely clever mathematicians. The disproof, by Florian Frick, is found in the paper Counterexamples to the Topological Tverberg Conjecture.

More Information

From Oberwolfach: The Topological Tverberg Conjecture is False, at Gil Kalai’s blog

Counterexamples to the Topological Tverberg Conjecture, by Florian Frick on the ArXiv

Florian Frick’s TU Berlin homepage

via Gil Kalai on Google+