Lie group actions with polytopal orbit space

• Applicant: Professor Dr. Oliver Goertsches
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 549576812

This project stems from the observation that the classes of compact manifolds with Lie group action 1. Toric symplectic manifolds / Quasitoric manifolds 2. Multiplicity-free Hamiltonian actions 3. Cohomogeneity one manifolds with a nonprincipal orbit 4. Hyperpolar actions on simply-connected symmetric spaces share similar properties: a) Their orbit space is homeomorphic to a polytope. b) They are determined by, or even reconstructible from the combinatorics of the orbit space, when equipped with appropriate (isotropy) data. c) Their equivariant cohomology (over the rationals respectively the reals) is computable, and algebraically as simple as possible: it is a Cohen-Macaulay module; in particular, if the action in question has an isotropy group of maximal rank, then it is a free module. We will investigate arbitrary actions of compact Lie groups on smooth manifolds whose orbit space is homeomorphic to a polytope, such that the orbit type strata correspond to the faces of the polytope -- we speak about actions with polytopal orbit space. The guiding question we pose is whether, in the same way as toric symplectic geometry has the theory of multiplicity-free Hamiltonian actions as a natural nonabelian analogue, in how far the more topological theory of quasitoric manifolds can be extended to the nonabelian setting. To this end, we pursue several topological and geometrical goals. We wish to understand in how far actions with polytopal orbit space are determined by the combinatorics of the labelled orbit space, we will put this condition into context with that of polarity of the action, and compute equivariant and non-equivariant topological invariants of these actions, like the (equivariant) cohomology ring.

• DFG Programme: Research Grants