Tverberg's Transversal Conjecture and Analogues of Nonembeddability Theorems for Transversals

In this paper we prove a special case of the transversal conjecture of Tverberg and Vrećica. We consider the case when the numbers of parts ri in this conjecture are powers of the same prime. We also prove some results on common transversals that generalize the classical nonembeddability theorems. We also prove an analogue of colored Tverberg’s theorem by Živaljević and Vrećica. Instead of multicolor simplices with common point it gives multicolor simplices with common m-transversal. 1. Tverberg’s transversal conjecture In this paper we prove a special case of the transversal conjecture of Tverberg and Vrećica. Conjecture 1. Let 0 ≤ m ≤ d − 1 and let S0, S1, . . . , Sm be m + 1 finite sets in R. Let |Si| = (ri − 1)(d − m + 1) + 1. Then every set Si can be partitioned into ri parts Si1, Si2, . . . , Siri so that all the sets convSij can be met by a single m-flat. This conjecture was formulated by H. Tverberg on the 1989 Symposium on Combinatorics and Geometry in Stockholm. In print it was first formulated by H. Tverberg and S. Vrećica in [7], where a special case of this conjecture was proved. In the papers [11, 12] R. Živaljević and S. Vrećica established the case of this conjecture when all ri are equal to the same prime p, if p is odd then d and m were also required to be odd. Here we prove the theorem that generalizes the results of R. Živaljević and S. Vrećica to prime powers and show that the condition that d is odd is not necessary. Besides, our proof is quite short because of using the multiplicative rule for the Euler class. We prove that this conjecture is true when the numbers ri are powers of the same prime ri = p ki , and for odd p, we also need d−m to be even. Similarly to what was done by R. Živaljević and S. Vrećica, we prove a more general topological version of this conjecture. Theorem 1. Let 0 ≤ m ≤ d−1, let ri (i = 0, . . . ,m) be powers of the same prime ri = pi. If p is odd, let d−m be even. Let for each i = 0, . . . ,m fi map continuously an (ri−1)(d−m+1)-dimensional simplex ∆i = ∆ (ri−1)(d−m+1) to R. This research was supported by the Russian Foundation for Basic Research grant No. 06-01-00648, and by the President’s of Russian Federation grant No. MK-5724.2006.1. 1