Luis Casian

The paper concerns the topology of an isospectral real smooth manifold for certain Jacobi element associated with real split semisimple Lie algebra. The manifold is identified as a compact, connected completion of the disconnected Cartan subgroup of the corresponding Lie group $\tilde G$ which is a disjoint union of the split Cartan subgroups associated to semisimple portions of Levi factors of all standard parabolic subgroups of $\tilde G$. The manifold is also related to the compactified level sets of a generalized Toda lattice equation defined on the semisimple Lie algebra, which is diffeomorphic to a toric variety in the flag manifold ${\tilde G}/B$ with Borel subgroup $B$ of $\tilde G$. We then give a cellular decomposition and the associated chain complex of the manifold by introducing colored-signed Dynkin diagrams which parametrize the cells in the decomposition.

This paper concerns the topology of isospectral real manifolds of certain Jacobi elements associated with real split semisimple Lie algebras. The manifolds are related to the compactified level sets of the generalized (non-periodic) Toda lattice equations defined on the semisimple Lie algebras. We then give a cellular decomposition and the associated chain complex of the manifold by introducing coloured Dynkin diagrams which parametrize the cells in the decomposition. We also discuss the Morse chain complex of the manifold.

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.

Although these compactified varieties are singular, they resemble certain smooth Schubert varieties e.g. they both have a cell decomposition consiting of unipotent group orbits of the same dimensions. In particular, for the case of a Lie algebra of type $A$ the rational homology/cohomology obtained from the compactified isospectral variety of the nilpotent Toda lattice equals that of the corresponding Schubert variety.

We consider a realization of the real Grassmann manifold Gr(k,n) based on a particular flow defined by the corresponding (singular) solution of the KP equation. Then we show that the KP flow can provide an explicit and simple construction of the incidence graph for the integral cohomology of Gr(k,n). It turns out that there are two types of graphs, one for the trivial coefficients and other for the twisted coefficients, and they correspond to the homology groups of the orientable and non-orientable cases of Gr(k,n) via the Poincare-Lefschetz duality. We also derive an explicit formula of the Poincare polynomial for Gr(k,n) and show that the Poincare polynomial is also related to the number of points on a suitable version of Gr(k,n) over a finite field $\F_q$ with q being a power of a prime. In particular, we find that the number of $\F_q$ points on Gr(k,n) can be computed by counting the number of singularities along the KP flow.

This paper concerns the topology of isospectral real manifolds of certain Jacobi elements associated with real split semisimple Lie algebras. The manifolds are related to the compactified level sets of the generalized (nonperiodic) Toda lattice equations defined on the semisimple Lie algebras. We then give a cellular decomposition and the associated chain complex of the manifold by introducing colored Dynkin diagrams which parametrize the cells in the decomposition. We also discuss the Morse chain complex of the manifold.