# 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 integrartion 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-connected coverings

6.5 Reducible spaces

**7. Nonvanishing Euler Characteristics**

**8. Classification until Dimension 6**

**9. Classification in Dimension 7**

**10. Classification in Dimension 8**

**11. Semisimple Spaces**

### 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, D
^{n}, S^{n-1}, e^{n}open n-cell - CW complex, n-skeleton Sk
^{n}(X), attaching maps, properties "CW", cellular map - CW complex of finite type, c
_{t}(X), Euler characteristics, two structures for S^{n} - Sum XuY, product XxY, quotient X/A, wedge sum XvY, smash product X∧Y
- Reduced/unreduced cone CX and suspension SX,
- Identifications S
^{n}= I^{n}/∂I^{n}and S^{n+m}= S^{n}∧ S^{m }

1.2 Projective and Grassmann spaces

- Projective spaces FP
^{n}for F = R, C and H (Hamiltonians) - FP
^{n}= FP^{n-1}u e^{dn }and CW-structures - Octonions O, octonion projective line OP
^{1}and plane OP^{2} - 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 C
_{0}(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 π
_{1}(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, G
_{2}and F_{4}

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**

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

- Abstract and concrete homogeneous spaces
- Homogeneous representations
- Reducible representations

6.2 Homogeneous fibre bundles

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

6.4 Fundamental group, universal covering and simply connected spaces

6.5 Semisimple spaces and 1-connected coverings

6.5 Reducible spaces

**7. Nonvanishing Euler Characteristics**

**8. Classification until Dimension 6**

**9. Classification in Dimension 7**

**10. Classification in Dimension 8**

**11. Semisimple Spaces**

### 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