Homology, Homotopy and Applications

Volume 24 (2022)

Number 2

Optimal colored Tverberg theorems for prime powers

Pages: 69 – 92

DOI: https://dx.doi.org/10.4310/HHA.2022.v24.n2.a4

Authors

Duško Jojić (Faculty of Science, University of Banja Luka, Republic of Srpska, Bosnia and Herzegovina)

Gaiane Panina (Department of Mathematics and Computer Science, St. Petersburg State University, St. Petersburg, Russia; and Steklov Mathematical Institute, St. Petersburg, Russia)

Rade Živaljević (Mathematical Institute, Serbian Academy of Sciences and Arts, Belgrade, Serbia)

Abstract

The colored Tverberg theorem of Blagojević Matschke, and Ziegler (Theorem 1.4) provides optimal bounds for the colored Tverberg problem, under the condition that the number of intersecting rainbow simplices $r = p$ is a prime number.

Our Theorem 1.6 extends this result to an optimal colored Tverberg theorem for multisets of colored points, which is valid for each prime power $r = p^k$, and includes Theorem 1.4 as a special case for $k = 1$. 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.

A quite different generalization arises if we consider colored classes that are (approximately) two times smaller than in the classical colored Tverberg theorem. Theorem 1.8, which unifies and extends some earlier results of this type, is based on the constraint method and uses the high connectivity of the configuration space.

Keywords

Tverberg theorem, chessboard complex, degree of equivariant maps, configuration space/test map scheme

2010 Mathematics Subject Classification

05E45, 52A35, 52C99, 55M20, 55M25, 55M35, 57S25

Full Text (PDF format)

Received 28 February 2021

Received revised 3 July 2021

Accepted 5 July 2021

Published 10 August 2022