Detalhes
TOPOLOGIA DIFERENCIAL II
Nome da Disciplina: TOPOLOGIA DIFERENCIAL II
Carga Horária: 60
Créditos: 4
Disciplina Regular: Sim
EMENTA
1. Lie groups and Lie algebra actions on manifolds
- Lie groups and Lie algebras: definitions, examples. Exponential map. Statement of integrability results.
- Actions of Lie groups and Lie algebras on manifolds: definition, examples (guiding example throughout the course: adjoint action of Lie group on its Lie algebra). Basic invariants: isotropy Lie group/Lie algebra and isotropy representation, orbit, orbit space. Orbits of Lie group actions are initial submanifolds. Integrability of Lie algebra actions.
- Proper actions: orbit spaces are Hausdorff.
2. Proper Lie group actions on manifolds
- Free and proper actions: principal bundles, statement of Godement's criterion for quotients, equivalence between free and proper actions of Lie groups on manifolds and principal bundles. Examples: projective spaces, flag manifolds, Stiefel manifolds.
- Linearisation results: Bochner linearisation theorem, existence of slices, tube theorem.
3. Differential topology of orbit spaces of proper actions
- Smooth structure via invariant functions, existence of invariant Riemannian metrics. Schwarz's theorem on invariant functions*.
- Orbit type stratification theorem: definition of orbit types, definition of Whitney stratification. Orbit types give a Whitney stratification of the orbit space.
- Regular and principal orbits: the principal orbit theorem.
- Desingularisation of proper actions: blowing up.
4. Application: compact Lie groups
- Develop as much of the theory of compact Lie groups and compact Lie algebras using the tools developed in the course.
*Time permitting.
- Lie groups and Lie algebras: definitions, examples. Exponential map. Statement of integrability results.
- Actions of Lie groups and Lie algebras on manifolds: definition, examples (guiding example throughout the course: adjoint action of Lie group on its Lie algebra). Basic invariants: isotropy Lie group/Lie algebra and isotropy representation, orbit, orbit space. Orbits of Lie group actions are initial submanifolds. Integrability of Lie algebra actions.
- Proper actions: orbit spaces are Hausdorff.
2. Proper Lie group actions on manifolds
- Free and proper actions: principal bundles, statement of Godement's criterion for quotients, equivalence between free and proper actions of Lie groups on manifolds and principal bundles. Examples: projective spaces, flag manifolds, Stiefel manifolds.
- Linearisation results: Bochner linearisation theorem, existence of slices, tube theorem.
3. Differential topology of orbit spaces of proper actions
- Smooth structure via invariant functions, existence of invariant Riemannian metrics. Schwarz's theorem on invariant functions*.
- Orbit type stratification theorem: definition of orbit types, definition of Whitney stratification. Orbit types give a Whitney stratification of the orbit space.
- Regular and principal orbits: the principal orbit theorem.
- Desingularisation of proper actions: blowing up.
4. Application: compact Lie groups
- Develop as much of the theory of compact Lie groups and compact Lie algebras using the tools developed in the course.
*Time permitting.
BIBLIOGRAFIA
J.J. Duistermaat and J.A.C. Kolk, Lie groups, Springer-Verlag, Berlin, Heidelberg, 2000.
G.W. Schwarz, Smooth functions invariant under the action of a compact Lie group, Topology 14 (1975), 63 -- 68.
G.W. Schwarz, Smooth functions invariant under the action of a compact Lie group, Topology 14 (1975), 63 -- 68.
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