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.

TOC, Topics and keywords on the content (preliminary)

Table of contents (preliminary)

Part I: Basic Material

1. Basic Homotopy Theory

1.1 CW complexes
1.2 Projective and Grassmann 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 and products
2.3 Thom isomorphism and Gysin LES
2.4 Characteristic classes
2.5 Serre spectral sequence

3. Smooth Manifolds

3.1 Smooth manifolds
3.2 Vector Fields and flows
3.3 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+1/2)-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

TOC with topics and keywords on the content (preliminary)

Part I: Basic Material

1. Basic Homotopy Theory

1.1 CW complexes

Basic point set topology assumed to be known
Basic spaces: I, Dⁿ, S^n-1, eⁿ open n-cell
CW complex, n-skeleton Skⁿ(X), attaching maps, properties "CW", cellular map
CW complex of finite type, c_t(X), Euler characteristics, two structures for Sⁿ
Sum XuY, product XxY, quotient X/A, wedge sum XvY, smash product X∧Y
Reduced/unreduced cone CX and suspension SX,
Identifications Sⁿ = Iⁿ/∂Iⁿ and S^n+m = Sⁿ ∧ S^m

1.2 Projective and Grassmann spaces

Projective spaces FPⁿ for F = R, C and H (Hamiltonians)
FPⁿ = FP^n-1 u e^dnand CW-structures
Octonions O, octonion projective line OP¹ and plane OP²
Grassmann spaces
Stiefel spaces
Remark: CW structures for these spaces (Schubert cells)

1.3 Mapping spaces

Mapping spaces C(X,Y) and pointed mapping space C₀(X,Y)
Path space PX and loop space ΩX
Exponential laws and adjunctions C(CX,Y)=C(X,PY), C(SX,Y)=C(X,ΩY)
Category CGWH of compactly generated weak Hausdorff spaces

1.4 Homotopy and homotopy groups

Homotopy is an equivalence relation, free and pointed versions [X,Y]^free and [X,Y]
Functoriality, respected by composition
Homotopy equivalence (h.e.) of maps and spaces
Induces bijections for [Z, ] and [ ,Z], contractible space
Homotopy groups π_n(X), are abelian for n>1, functor
Adjunction π_n(ΩX) = π_n+1(X)
Cellular approximation, application to π_n(S^m)
Thm of Milnor for C(X,Y)

1.5 Fibrations and fibre bundles

Homotopy lifting property, fibration (in the sense of Hurewitz or Serre)
Fibre, converting a map into a fibration
Mapping fibre sequence
Fibration induces a long exact sequence (LES) in π_*
Fibre bundle (FB) and principal fibre bundle (PFB)
Example: coverings, classification by fundamental group π₁(X)
Hopf maps
Vector bundle and associated sphere bundle
Theorem: a FB is a fibration
Universal PFB
Classifying space BG and classification theorem

1.6 Cofibrations

Homotopy extension property, cofibration
Cofibre, converting a map into a cofibration
Closed cofibration and mapping cone
Puppe cofibre sequence

2. Basic Homology Theory

2.1 Homology for CW complexes

Axioms of homology
Existence: singular homology, McCord homology
Consequences: suspensions, spheres and Mayer-Vietoris LES
Cellular homology
Homology of projective spaces and Grassmann spaces
Universal coefficient theorem
Künneth theorem
Hurewicz theorem
Homology spheres

2.2 Cohomology and products

Axioms of cohomology and existence
Products
Universal coefficient theorem
Künneth theorem
Suspension SX has trivial products

2.3 Thom isomorphism and Gysin LES

Thom class and Thom isomorphism
Gysin LES
Cohomology ring of projective spaces
Iterated sphere bundles
Homology spheres
Examples: classical groups and homogeneous spaces

2.4 Characteristic classes

General characteristic classes for PFB
Stiefel-Whitney classes: axioms and existence
Chern classes: axioms and existence
Euler class: axioms and existence
Pontrjagin classes: axioms and existence
Cohomology ring of Grassmann spaces

2.5 Serre spectral sequence

What is a spectral sequence (SPS)?
Serre SPS
Some immediate consequences
Gysin LES and Wang LES
Collapse theorem of Leray-Hirsch
Borel-Serre Theorem

3. Smooth Manifolds

3.1 Smooth manifolds

Manifolds, chart transitions in an atlas, smoothness
Smooth maps, diffeomorphisms and embeddings
Tangent space, tangent bundle and derivative of a smooth map
Manifolds with boundary
Regular values, theorem of Sard, transversality
Oriented manifolds
Mapping degree

3.2 Vector Fields and flows

Tangent vector fields as first order ODE systems
Local and global flows
Completeness for compact manifolds
Exponential map
Lie derivative

3.3 Differential Forms

Differential forms, exterior product and differential
Pull back and naturality
Orientation and integration
DeRham cohomology
DeRham theorem
Fundamental class and mapping degree

3.4 Poincaré duality

Poincaré duality
Poincaré-Lefschetz duality
Intersection form and signature
Remark on symmetric bilinear forms
Remark on torsion part (linking form)

3.5 Bordism

Bordism and zero-bordism
Unoriented and oriented bordism rings
Characteristic numbers as obstructions to zero-bordism
Examples: projective spaces
Theorem of Thom: unoriented bordism ring
Theorem of Thom: rational oriented bordism ring
Theorem of Wall (oriented bordism)
Remarks on further bordism groups

4. Compact Lie Groups

4.1 Lie groups and Lie algebras

Lie groups
Lie subgroups and the closed subgroup theorem of Cartan
Connected component and component group
Covering groups
Compact Lie groups and Malcev-Iwasawa splitting
Classical examples of Lie groups
Lie algebra of a Lie group
Canonical parallelization
Exponential map
Remark: Baker-Campbell-Hausdorff formula
Remark: Theorem of Ando
Adjoint representation
Classical examples of Lie algebras
G2 and F4

4.2 Representations of compact groups

Haar integral for compact topological groups
Example: invariant integration for compact Lie groups
Complex representations, characters and class functions
Orthogonality relations
Tensor products
Representation ring
Remark: Peter-Weyl theorem

4.3 Compact Lie groups and maximal tori

Classification of abelian Lie groups
Main theorem on maximal tori
Rank of a compact Lie group
Weyl group
Weights of a representation

4.4 Root systems and classification of simple Lie groups

Roots: global, local, infinitesimal
Root systems
Action of the Weyl group
Weyl chambers and simple roots
Dynkin diagram
Simple and semisimple groups
Compact groups
Killing-Cartan classification
Low dimensional isomorphisms
Tables for the classical groups, G₂ and F₄

4.5 Invariant integrartion and applications

Invariant integration
Fundamental group
The second homotopy group vanishes
Remark on higher homotopy groups

4.5 Complex representations

Highest weight theorem
Weyl character formula
Weyl dimension formula
Reduction of tensor products and formulas of Kostant
Remark: Borel-Weil theorem

4.6 Real and quaternionic representations

5. Riemannian Manifolds

PART II: Core Material

6. Compact Homogeneous Spaces

6.1 Smooth homogeneous spaces

Homogeneous spaces and Lie pairs
Connectedness
Compactness
Standard examples
Some low dimensional diffeomorphisms

6.2 Homogeneous fibre bundles

Homogeneous tangent bundle
Homogeneous vector and fibre bundles
Chains of subgroups and towers of homogeneous fibre bundles

6.4 Fundamental group, universal covering and simply connected spaces

Fundamental group of a Lie pair
Covering of a Lie pair
Simply connected Lie pairs

6.5 Semisimple spaces and (1+1/2)-connected coverings

Semisimple Lie pairs
Torus bundles
Second homotopy group and (1+1/2)-connected coverings
Semisimple homogeneous spaces

6.5 Construction of all irreducible Lie pairs

Reducible Lie pairs
Type I and II Lie pairs
Maximal subgroups and the gap theorem of Mann
Systematic construction

7. Nonvanishing Euler Characteristics

7.1 Flag manifolds

A Morse function for G/T
Cohomology ring of G/T
Euler characteristics and Weyl group

7.2 The case of equal rank

Euler characteristics and Weyl groups

7.3 The case of different rank

Vanishing of the Euler characteristics

8. Classification until Dimension 6

8.1 Classification in dimension 2, 3 and 4

8.2 Classification in dimension 5

Properties of the Wu manifold SU₃/SO₃

8.3 Classification in dimension 6

Properties of the flag space SU₃/T²

9. Classification in Dimension 7

9.1 Irreducible Lie pairs in dimension 7

9.2 The Aloff-Wallach series

Properties of the spaces SU₃/Δ(U₁)

9.3 The Witten series

Properties of the spaces SU₃xSU₂/Δ(SU₂xU₁)

9.4 The Bermbach series

Properties of the spaces SU₂xSU₂xSU₂/Δ(U₁xU₁)

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

Properties of the Berger manifold Sp₂/Δ(SU₂)

10.3 The Klaus series

Properties of the spaces SU₂xSU₂xSU₂/Δ(U₁)

10.4 Generalized Klaus manifolds

Properties of the spaces (SU₂)ⁿ/Δ(U₁)

10.5 Explicit classification in dimension 8

Theorem of Klaus: There are exactly 20 spaces up to diffeomorphy

10.6 On the spaces beyond dimension 8

Existence of infinitely many spaces

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