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On deformations of foliations and characteristic classes

Taro Asuke

Abstract

We study characteristic classes for deformations of foliations. Those classes include known classes such as the Godbillon--Vey class and the Fuks--Lodder--Kotschick class. We introduce a certain differential graded algebra (DGA for short) which recovers the Bott vanishing and some formulae by Heitsch. Some basic properties and structures of the cohomology of those DGA's are discussed. In particular, it is shown that at the level of the cohomology of DGA, there are some classes which cannot be described by the Godbillon--Vey class and the Fuks--Lodder--Kotschick class. It is also shown that if a certain type of characteristic classes admit non-trivial deformations in examples, then they yield another kind of classes which admit also non-trivial deformations.

On deformations of foliations and characteristic classes

Abstract

We study characteristic classes for deformations of foliations. Those classes include known classes such as the Godbillon--Vey class and the Fuks--Lodder--Kotschick class. We introduce a certain differential graded algebra (DGA for short) which recovers the Bott vanishing and some formulae by Heitsch. Some basic properties and structures of the cohomology of those DGA's are discussed. In particular, it is shown that at the level of the cohomology of DGA, there are some classes which cannot be described by the Godbillon--Vey class and the Fuks--Lodder--Kotschick class. It is also shown that if a certain type of characteristic classes admit non-trivial deformations in examples, then they yield another kind of classes which admit also non-trivial deformations.
Paper Structure (10 sections, 61 theorems, 58 equations)

This paper contains 10 sections, 61 theorems, 58 equations.

Table of Contents

  1. Definitions
  2. Normal bundles of higher order
  3. Spaces of deformations
  4. Basic definitions
  5. Relevant differential graded algebras
  6. Bott vanishing theorem and Heitsch formula
  7. Characteristic mapping
  8. Variability and rigidity, comparison of $\chi$ and $\chi'$
  9. Structure of $H^*(\mathrm{D}^\infty\mathrm{W}_q)$
  10. A partial description of $H^*(\mathrm{D}^\infty\mathrm{W}_1)$

Key Result

Theorem 2.14

If we set $H^{(k)}=\tau^{r-k}H^{(r)}$ for $0\leq k\leq r$, then we have $N^r\cong\bigoplus_{k=0}^rH^{(k)}$. Moreover, each $H^{(k)}$ is isomorphic to $Q'=\widetilde{p}^r{}^*Q$.

Theorems & Definitions (175)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • ...and 165 more