On Voronoi’s two tilings of the cone of metrical forms

In his last two papers Georges Voronoi introduced two tilings for the cone of metrical forms for lattices, the tiling by perfect domains and the tiling by lattice type domains. Both are facet-to-facet tilings by polyhedral subcones, and invariant with respect to the natural action of GL(n, Z) on Sym(n, R). In working out the details of his theory of lattice types, his most important legacy for geometers, Voronoi observed that the tiling by lattice types refines the tiling by perfect domains in the case of 2-, 3and 4-dimensional lattices. The technical advantage gained in this way allowed him to work out the details of his classification theory for four-dimensional lattices. The work of all researchers following Voronoi relied heavily on this hypothesis, that the tiling by lattice types refines the tiling by perfect domains. In particular, the classification of the 5-dimensional generic lattice types by Ryshkov and Baranovski would not have been possible without invoking this hypothesis. In this note we show that this hypothesis does not hold in six dimensions, and therefore in all higher dimensions. This paper is an announcement of results and will be followed by a more complete treatment where all proofs are given. This paper is dedicated to Peter Gruber on the occassion of his sixtieth birthday.