Homology, Homotopy and Applications
Volume 24 (2022)
Optimal colored Tverberg theorems for prime powers
Pages: 69 – 92
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.
Tverberg theorem, chessboard complex, degree of equivariant maps, configuration space/test map scheme
2010 Mathematics Subject Classification
05E45, 52A35, 52C99, 55M20, 55M25, 55M35, 57S25
Received 28 February 2021
Received revised 3 July 2021
Accepted 5 July 2021
Published 10 August 2022