Optimal colored Tverberg theorems for prime powers

TL;DR

This paper extends the optimal colored Tverberg theorem to prime power cases by using smaller simplices and multiple vertex appearances, broadening the theorem's applicability.

Contribution

It introduces an extension of the colored Tverberg theorem to prime powers, utilizing abridged simplices and multiple vertex repetitions for broader validity.

Findings

Extended theorem to prime powers $r=p^k$

Used abridged simplices to reduce dimension

Applied Eilenberg-Krasnoselskii degree theory

Abstract

The type A colored Tverberg theorem of Blagojevi\'{c}, Matschke, and Ziegler provides optimal bounds for the colored Tverberg problem, under the condition that the number of intersecting rainbow simplices is a prime number. We extend this result to an optimal, type A colored Tverberg theorem for multisets of colored points, which is valid for each prime power $r=p^k$. One of the principal new ideas is to replace the ambient simplex $\Delta^N$, used in the original Tverberg theorem, by an "abridged simplex" of smaller dimension, and to compensate for this reduction by allowing vertices to repeatedly appear a controlled number of times in different rainbow simplices. Configuration spaces, used in the proof, are combinatorial pseudomanifolds which can be represented as multiple chessboard complexes. Our main topological tool is the Eilenberg-Krasnoselskii theory of degrees of equivariant maps for non-free actions.