A tight colored Tverberg theorem for maps to manifolds

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We prove that any continuous map of an N-dimensional simplex Deltaₙ with colored vertices to a d-dimensional manifold M must map r points from disjoint rainbow faces of Deltaₙ to the same point in M: For this we have to assume that N ≥ (r-1)(d+1), no r vertices of Deltaₙ get the same color, and our proof needs that r is a prime. A face of Deltaₙ 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=Rᵈ. 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.