The topology of Gelfand-Zeitlin fibers

We prove several new results about the topology of fibers of Gelfand-Zeitlin systems on unitary and orthogonal coadjoint orbits. First, we provide a simplified description of their diffeomorphism types as balanced products, recovering results of Bouloc-Miranda-Zung as special cases. Second, we compute their cohomology rings. Finally, we complete the computation of their first three homotopy groups (the first and second homotopy groups were described by Cho-Kim-Oh in the unitary case). Our results build on Cho-Kim-Oh's description of Gelfand-Zeitlin fibers as total spaces of certain towers of fiber bundles. Our description of the diffeomorphism type, cohomology ring, and homotopy groups can be read in a straightforward manner from the combinatorics of the associated Gelfand-Zeitlin pattern.