Geometric Theory of Foliations

Name: Geometric Theory of Foliations

Author(s): Alcides Lins Neto e César Camacho

Pages: 244

Publication: IMPA, 2019

ISBN: 978-85-244-0452-8

Edition: 2

1. Differentiable varieties
1.1 Differentiable Varieties
1.2 The derivative
1.3 Immersions and submersions
1.4 Subvarieties
1.5 Regular Values
1.6 Transversality
1.7 Unit Partitioning

2. Foliations
2.1 Foliations
2.2 The leaves
2.3 Distinguished Applications
2.4 Plane and Foliation Fields
2.5 Guidance
2.6 Swivel double coating
2.7 Orientable and Transversely Orientable Foliations
Notes to Chapter 2
Exercises

3. Leaf Topology
3.1 Spacing of the sheets
3.2 Transverse uniformity
3.3 Closed leaves
3.4 Minimal Sets of Laminations
Notes to Chapter 3
Exercises

4 Holonomy and stability theorems
4.1 Holonomy of a sheet
4.2 Determination of the germ of a leaf in the vicinity of a leaf by the holonomy of the leaf
4.3 Motto of global trivialization
4.4 Reeb's local stability theorem
4.5 Complex Stability Theorem. Transversely Orientable Case
4.6 Complete Stability Theorem. General Case
Chapter 4 Notes
Exercises

Fiberboard and veneers
5.1 Fiber optic spaces
5.2 Transverse foliations across the fibers of a fiberboard
5.3 The holonomy of F
5.4 Suspension of a presentation
5.5 Presence of leaf germs
5.6 Sacksteder Example
Notes to Chapter 5
Exercises

6 Analytical foliations of codimension one
6.1 About Theorem 6.1
6.2 Singularities of the mappings ƒ : R ⁿ → R
6.3 Haefliter's Construction
6.4 Foliations with singularities in ^D2
6.5 Proof of Haefliger's theorem
Exercises

7 Novikov's Theorem
7.1 Demonstration Outline
7.2 Vanishing Cycles
7.3 Simple vanishing cycles
7.4 Existence of the compact sheet
7.5 Existence of the Reeb component
7.6 Other results from Novikov
7.7 The non-guidable case
Exercises

8 Topological Aspects of Group Action Theory
8.1 Elementary Properties
8.2 The rank theorem of S ³
8.3 Generalization of the rank theorem
8.4 Poincaré-Bendixson Theorem for Actions in ^R²
8.5 Actions of the group of affine transformations of the real line
Exercises

Fundamental Group and Covering Spaces
A.1 Homeotopies
A.2 Induced homomorphism
A.3 Spaces with the same type of homotopy
A.4 Calculation of the fundamental group of some varieties. Particular forms of van Kampen's theorem.
A.5 Coverage spaces
A.6 Universal coating
A.7 Overlay automorphisms

Frobenius' Theorem
B.1 Vector fields and Lie brackets
B.2 Frobenius Theorem
B.3 Plane fields defined by differential shapes