Introduction to the Geometry of Foliations, Part B: Foliations of Codimension One

ISBN-13: 9783528085681 / Niemiecki / Miękka / 1983 / 298 str.

Gilbert Hector

Spis treści

IV — Basic Constructions and Examples.- 1. General setting in co dimension one.- 1.1. Existence of a transverse foliation.- 1.2. Holonomy pseudogroups.- 1.3. Appendix: One-dimensional foliations and local flows.- 2. Topological dynamics.- 2.1. The relation ?F and ?P.- 2.2. Leaf types; minimal sets.- 3. Foliated bundles; examples.- 3.1. Topological dynamics in foliated bundles.- 3.2. Fibre bundles arising as foliated bundles.- 3.3. Examples.- 4. Gluing foliations together.- 4.1. Gluing together foliations tangent to the boundary.- 4.2. Gluing together foliations transverse to the boundary.- 5. Turbulization.- 5.1. Closed transversals.- 5.2. Turbulization along a closed transversal or along a boundary component.- 6. Codimension-one foliations on spheres.- 6.1. Manifolds as open books.- 6.2. Foliations on odd-dimensional spheres.- V — Structure of Codimension-One Foliations.- 1. Transverse orientability.- 1.1. Transverse orientability; one- and two-sided leaves.- 1.2. Forms and linear holonomy.- 2. Holonomy of compact leaves.- 2.1. Local diffeomorphisms of the real line.- 2.2. Germ near a compact leaf; local stability.- 3. Saturated open sets of compact manifolds.- 3.1. Semi-proper leaves; completion of saturated open sets.- 3.2. The structure of saturated open sets.- 4. Centre of a compact foliated manifold; global stability.- 4.1. Structure of the centre.- 4.2. The global stability theorems of Reeb and Thurston.- VI — Exceptional Minimal Sets of Compact Foliated Manifolds; A Theorem of Sacksteder.- 1. Resilient leaves.- 2. The theorem of Denjoy-Sacksteder.- 3. Sacksteder’s theorem.- 4. The theorem of Schwartz.- VII — One Sided Holonomy; Vanishing Cycles and Closed Transversals.- 1. Preliminaries on one-sided holonomy and vanishing cycles.- 2. Transverse foliation* of D2 × IR.- 2.1. Foliations with singularities on the disk.- 2.2. One-sided holonomy in transverse foliations.- 3. Existence of one-sided holonomy and vanishing cycles.- VIII — Foliations without Holonomy.- 1. Closed 1-forms without singularities.- 1.1. Closed 1-forms and foliations obtained by an equivariant fibration.- 1.2. The theorem of Tischler.- 2. Foliations without holonomy versus equivariant fibrations.- 2.1. Trivialization and global unwrapping.- 2.2. Trivializing foliations without kolonomy.- 3. Holonomy representation and cohomology direction.- 3.1. Holder’s theorem; fixed point free subgroups of Homeo (IR).- 3.2. Foliations witkout kolonomy and closed 1-forms.- IX — Growth.- 1. Growth of groups, homogeneous spaces and riemannian manifolds.- 1.1. Growth type, of functions.- 1.2. Growth of finitely generated groups and komogeneous spaces.- 1.3. Growth of riemannian manifolds; application to covering spaces.- 2. Growth of leave in foliatoons on compact manifolds.- 2.1. Growth of leaves in topological foliations.- 2.2. Growth of leaves in differentiable foliations.- X — Holonomy Invariant Measures.- 1. Invariant measures for subgroups of Homeo (?) or Homeo(S1).- 1.1. Abelianization of subgroups of Homeo+(IR) admitting an invariant measure.- 1.2. Diffuse measures versus Lebesgue measure; invariant measures on S1.- 2. Foliations with holonomy invariant measure.- 2.1. Fundamentals on holonomy invariant measures.- 2.2. Averaging sequences and kolonomy invariant measunres.- 2.3. Holonomy invariant measures for foliations of codimension one.