The Tverberg-Vrećica problem and the combinatorial geometry on vector bundles

Abstract

It is shown that many classical and many new combinatorial geometric results about finite sets of points in Rd, specially the theorems of Tverberg type, can be generalized to the case of vector bundles, where they become combinatorial geometric statements about finite families of continuous cross-sections. The well known Tverberg-Vrećica conjecture is interpreted as a result of this type and its partial solution is obtained with the aid of the parametrized, ideal-valued, cohomological index theory. In the same spirit, classical "nonembeddability" and "coincidence" results like K3,3 (rightwards arrow with stroke) R2 have higher dimensional analogues. A new ingredient is that the coincidence condition is often interpreted as the existence of a common affine k-dimensional transversal, which reduces to the classical case for k = 0.