Ebook: Introduction to the Geometry of Foliations, Part A: Foliations on Compact Surfaces, Fundamentals for Arbitrary Codimension, and Holonomy

I — Foliations on Compact Surfaces.- 1. Vector fields on surfaces.- 1.1. Examples of isolated singularities.- 1.2. The index of an isolated singularity.- 1.3. The theorem of Poincaré — Bohl — Hopf.- 1.4. Existence of non-singular vector fields.- 2. Foliations on surfaces.- 2.1. Motivating remarks.- 2.2. Definition of foliations and related notions.- 2.3. Orientability; relation with vector fields.- 2.4. The existence theorem of Poincaré-Kneser.- 3. Construction of foliations.- 3.1. Suspensions.- 3.2. Germs near circle leaves; leaf holonomy.- 3.3. Reeb components.- 3.4. Turbulization.- 3.5. Gluing foliations together.- 4. Classification of foliations on surfaces.- 4.1. Topological dynamics.- 4.2. Foliations on the annulus and on the Möbius band.- 4.3. Foliations on the torus and on the Klein bottle.- 5. Denjoy theory on the circle.- 5.1. The rotation number.- 5.2. Denjoy’s example.- 5.3. Denjoy’s theorem.- 6. Structural stability.- 6.1. Structural stability for diffeomorphisms of the interval and the circle.- 6.2. Structural stability for suspensions.- 6.3. Structural stability for foliations in general.- Chatter II — Fundamentals on Foliations.- 1. Foliated bundles.- 1.1. Preparatory material on fibre bundles.- 1.1. Suspensions of group actions.- 1.3. Foliated bundles.- 1.4. Equivariant submersions.- 2. Foliated manifolds.- 2.1. Definition of a foliation; related notions.- 2.2. Transversality; orientability.- 2.3. The tangent bundle of a foliation; Frobenius’ theorem.- 2.4. Pfaffian forms; Frobenius’ theorem (dual version).- 3. Examples of foliated manifolds.- 3.1. Foliations defined by locally free group actions.- 3.2. Foliations with a transverse structure..- III — Holonom.- 1. Foliated microbundles.- 1.1. Localization in foliated bundles.- 1.2. Generalities on foliated microbundles.- 1.3. Holonomy of foliated microbundles.- 2. Holonomy of leaves.- 2.1. Unwrapping of leaves; leaf holonomy.- 2.2. Holonomy and foliated cocycles; leaves without holonomy.- 3. Linear holonomy; Thurston’s stability theorem.- 3.1. Linear and infinitesimal holonomy.- 3.2. Thurston’s stability theorem.- Literature.- Glossary of notations.