A tight colored Tverberg theorem for maps to manifolds
A coloring of the vertices of a simplex \(\Delta\) is a partition of the vertex set into color classes. A face of \(\Delta\) is called a rainbow face if its vertices have different colors. The authors prove the following result: Let \(N = (d + 1)(r - 1)\), where \(d \geq 1\) and \(r \geq 2\) is a prime number. If \(M\) is a \(d\)-dimensional manifold and \(\Delta_N\) is an \(N\)-dimensional simplex whose vertices are colored such that all color classes are of size at most \(r - 1\) then for every continuous map \(f: \Delta_N \rightarrow M\) there exist \(r\) disjoint rainbow faces of \(\Delta_N\) whose images under \(f\) have a point in common.
deleted product/join configuration space
scientific article