L04 - Geometry and topology of compact homogeneous spaces - Material
Stephan Klaus and Wilderich Tuschmann plan to publish a monograph on the seminar in the series "Oberwolfach Seminars" at Springer-Birkhäuser.
Table of contents (preliminary)
Part I: Basic Material
1. Basic Homotopy Theory
1.1 CW complexes
1.2 Projective, Stiefel, Grassmann and flag spaces
1.3 Mapping spaces
1.4 Homotopy and homotopy groups
1.5 Fibrations and fibre bundles
1.6 Cofibrations
2. Basic Homology Theory
2.1 Homology for CW complexes
2.2 Cohomology, products and operations
2.3 Thom isomorphism, Gysin LES and applications
2.4 Characteristic classes
2.5 Serre spectral sequence
3. Smooth Manifolds
3.1 Smooth manifolds
3.2 Tangent bundle, vector fields and flows
3.3 Mapping degree
3.4 Differential Forms
3.4 Poincaré duality
3.5 Bordism
4. Compact Lie Groups
4.1 Lie groups and Lie algebras
4.2 Representations of compact groups
4.3 Compact Lie groups and maximal tori
4.4 Root systems and classification of simple Lie groups
4.5 Invariant integration and applications
4.5 Complex representations
4.6 Real and quaternionic representations
5. Riemannian Manifolds
5.1 Riemannian metrics, manifolds and submanifolds
5.2 Levi-Civita Connection (and Christoffel Symbols)
5.3 Curvature Tensor (and derived curvature quantities)
5.4 Invariant Metrics on Lie groups and homogeneous spaces (O'Neill formulas)
5.4 Vector Fields and Covariant Derivatives along Maps
5.5 Parallel Transport, Geodesics and Riemannian Exponential Map
5.6 Jacobi Fields and the First and Second Variation of Energy
5.7 Completeness and the Hopf-Rinow Theorem
5.8 The theorems of Bonnet-Myers, Synge and Cartan-Hadamard
5.9 Spaces and Moduli Spaces of Riemannian Metrics
PART II: Core Material
6. Compact Homogeneous Spaces
6.1 Smooth homogeneous spaces
6.2 Homogeneous fibre bundles
6.4 Fundamental group, universal covering and simply connected spaces
6.5 Semisimple spaces and 1.5-connected coverings
6.5 Construction of all irreducible Lie pairs
7. Nonvanishing Euler Characteristics
7.1 Flag manifolds
7.2 The case of equal rank
7.3 The case of different rank
8. Classification until Dimension 6
8.1 Classification in dimension 2, 3 and 4
8.2 Classification in dimension 5
8.3 Classification in dimension 6
9. Classification in Dimension 7
9.1 Irreducible Lie pairs in dimension 7
9.2 The Aloff-Wallach series
9.3 The Witten series
9.4 The Bermbach series
9.5 Explicit classification in dimension 7
10. Classification in Dimension 8
10.1 Irreducible Lie pairs in dimension 8
10.2 The Berger manifold
10.3 The Klaus series
10.4 Generalized Klaus manifolds
10.5 Explicit classification in dimension 8
10.6 On the spaces beyond dimension 8
11. Semisimple Spaces
11.1 Semisimple spaces until dimension 8
11.2 Semisimple spaces in dimension 9
11.3 Semisimple spaces in dimension 10
11.4 Semisimple spaces in dimension11
11.5 Semisimple spaces in dimension 12
PART III: Applications in Geometry and Topology
12. Homogeneous Spaces of Positive Curvature
12.1 The list
12.2 Idea of proof
13. Manifolds of Positive Curvature
13.1 The other known examples: biquotients and cohomogeneity one manifolds
13.2 Obstructions to pinched positive sectional curvature
14. Moduli Spaces of Nonnegative Curvature Metrics
14.1 Connectedness and Disconnectedness Results in Positive Scalar Curvature
14.2 The s-invariant
14.3 Examples
14.4 Perspectives