2014/07/15 by Kathleen Smith, Smith, Kathleen
Mathematics · #57 #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals #Symplectic Geometry (math.SG) #math.SG #msc:57
paper · pdf · doi:10.48550/arxiv.1407.4351
63 pages; 6 figures. Ph.D. thesis of Kathleen Smith, Department of Mathematics, University of Toronto, 2013
arxiv created 2014/07/15 · openalex publication_date 2014/07/15 · arxiv updated 2014/07/17 · openalex created_date 2017/01/06 · openalex updated_date 2026/07/28
In the early 1980s a landmark result was obtained by Atiyah and independently Guillemin and Sternberg: the image of the momentum map for a torus action on a compact symplectic manifold is a convex polyhedron. Atiyah's proof makes use of the fact that level sets of the momentum map are connected. These proofs work in the setting of finite-dimensional compact symplectic manifolds. One can ask how these results generalize. A well-known example of an infinite-dimensional symplectic manifold with a finite-dimensional torus action is the based loop group. Atiyah and Pressley proved convexity for this example, but not connectedness of level sets. A proof of connectedness of level sets for the based loop group was provided by Harada, Holm, Jeffrey and Mare in 2006. In this thesis we study Hilbert manifolds equipped with a strong symplectic structure and a finite-dimensional group action preserving the strong symplectic structure. We prove connectedness of regular generic level sets of the momentum map. We use this to prove convexity of the image of the momentum map.