Twisted Tomei manifolds and the Toda lattices

Abstract

This paper begins with an observation that the isospectral leaves of the signed Toda lattice as well as the Toda flow itself may be constructed from the Tomei manifolds by cutting and pasting along certain chamber walls inside a polytope. It is also observed through examples that although there is some freedom in this procedure of cutting and pasting the manifold and the flow, the choices that can be made are not arbitrary. We proceed to describe a procedure that begins with an action of the Weyl group on a set of signs; it uses the Convexity Theorem in \cite{BFR:90} and combines the resulting polytope with the chosen Weyl group action to paste together a compact manifold. This manifold which is obtained carries an action of the Weyl group and a Toda lattice flow which is related to this action. This construction gives rise to a large family of compact manifolds which is parametrized by twisted sign actions of the Weyl group. For example, the trivial action gives rise to Tomei manifolds and the standard action of the Weyl group on the connected components of a split Cartan subgroup of a split semisimple real Lie group gives rise to the isospectral leaves of the signed Toda lattice. This clarifies the connection between the polytope in the Convexity Theorem and the topology of the compact smooth manifolds arising from the isospectral leaves of a Toda flow. Furthermore, this allows us to give a uniform treatment to two very different cases that have been studied extensively in the literature producing new cases to look at. Finally we describe the unstable manifolds of the Toda flow for these more general manifolds and determine which of these give rise to cycles.