Contents

• Foliations
  • What is a foliation and why is it interesting ?
  • Equivalent definitions
  • Holonomy grupoid
  • How to handle ``$M/\sF$''
  • Characteristic classes
  
  
• Characteristic classes
  • Preamble: Chern-Weil construction of Pontryagin ring
  • Adapted connection and Bott's theorem
  • The Godbillon-Vey class
  • Nontriviality of Godbillon-Vey class
  • Naturality under transversality
  • Transgressed classes
  
  
• Weil algebras
  • The truncated Weil algebras and characteristic homomorphism
  • $W_q$ and framed foliations
  
  
• Gelfand-Fuks cohomology
  • Cohomology of Lie algebras
  • Gelfand-Fuks cohomology
  • Some "soft" results
  • Spectral sequences
    • Exact couples
    • Filtered complexes
    • Illustration of convergence
    • Hochschild-Serre spectral sequence
  
  
• Characteristic maps and Gelfand-Fuks cohomology
  • Jet groups
  • Jet bundles
  • Characteristic map for foliation
  
  
• Index theory and noncommutative geometry
  • Classical index theorems
  • General formulation and proto-index formula
  • Multilinear reformulation: cyclic cohomology (Connes)
  • Connes cyclic cohomology
  • An alternate route, via the Families Index Theorem
  • Index theory for foliations
  
  
• Hopf cyclic cohomology
  • Preliminaries
    • Cyclic cohomology in abelian category
    • Hopf algebras
    • Motivation for Hopf-cyclic cohomology
    • Hopf-cyclic cohomology with coefficients
    • Special cases
  • The Hopf algebra $\sH_n$