  We prove that any continuous map of an N-dimensional simplex Delta_N with
colored vertices to a d-dimensional manifold M must map r points from disjoint
rainbow faces of Delta_N to the same point in M: For this we have to assume
that N \geq (r-1)(d+1), no r vertices of Delta_N get the same color, and our
proof needs that r is a prime. A face of Delta_N is a rainbow face if all
vertices have different colors.
  This result is an extension of our recent "new colored Tverberg theorem", the
special case of M=R^d. It is also a generalization of Volovikov's 1996
topological Tverberg theorem for maps to manifolds, which arises when all color
classes have size 1 (i.e., without color constraints); for this special case
Volovikov's proof, as well as ours, work when r is a prime power.
