Diffeological and differential spaces are generalisations of smooth structures on manifolds. We show that the “intersection” of these two categories is isomorphic to Frölicher
spaces, another generalisation of smooth structures. We then give examples of such spaces,
as well as examples of diffeological and differential spaces that do not fall into this category.
We apply the theory of diffeological spaces to differential forms on a geometric quotient
of a compact Lie group. We show that the subcomplex of basic forms is isomorphic to
the complex of diffeological forms on the geometric quotient. We apply this to symplectic
quotients coming from a regular value of the momentum map, and show that diffeological
forms on this quotient are isomorphic as a complex to Sjamaar differential forms. We
also compare diffeological forms to those on orbifolds, and show that they are isomorphic
complexes as well.
We apply the theory of differential spaces to subcartesian spaces equipped with families
of vector fields. We use this theory to show that smooth stratified spaces form a full
subcategory of subcartesian spaces equipped with families of vector fields. We give families
of vector fields that induce the orbit-type stratifications induced by a Lie group action, as
well as the orbit-type stratifications induced by a Hamiltonian group action.
| Identifer | oai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:OTU.1807/34959 |
| Date | 08 January 2013 |
| Creators | Watts, Jordan |
| Contributors | Karshon, Yael |
| Source Sets | Library and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada |
| Language | en_ca |
| Detected Language | English |
| Type | Thesis |
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