Colored Tverberg problem — history and recent developments

M. Gromov proposed a program of evaluating the complexity of the topology of the fibers $f^{-1}(y)\subseteq X$ of continuous maps $f: X \rightarrow Y$, in terms of combinatorial invariants of certain polyhedra and/or the cohomology algebras $H^\ast(X)$.

In the special case when $X$ is a simplicial complex and $Y$ an Euclidean space, a special attention has been given to the Topological Tverberg problem and its relatives. We plan to report on some recent advances in this area, focusing to the so called “Colored Tverberg problem”.