Godbillon-Vey invariants for families of foliations

Godbillon-Vey classes of a family of regular foliations

The aim of the paper is to construct some Godbillon-Vey classes of a family of regular foliations, defined in the paper. These classes are cohomology classes on the manifold or on suitable open subsets. Some examples are also considered.

Bott Integrability and Higher Integrability; Higher Cheeger-Simons and Godbillon-Vey Invariants

This paper studies the interaction of $π_1(M)$ for a $C^\infty$ manifold $M$ with Bott's original obstruction to integrability, and with differential geometric invariants such as Godbillon-Vey and Cheeger-Simons invariants of a foliation. We prove that the ring of higher Pontrjagin and higher Chern classes of an integrable subbundle $E$ of the tangent bundle of a manifold vanishes above dimension $2k$ where $k=dim(TM/E)$, and where the higher Pontrjagin and Chern rings are rings generated by $i^*y \cup p_j(TM/E)$ and by $i^*y \cup c_j(TM/E)$ respectively, with $p_j$ the $j$-th Pontrjagin class…

关于叶形的变形和特征类

On deformations of foliations and characteristic classes

本文研究叶形变形的特征类，引入一种微分格拉斯曼代数，讨论其上同调的性质，并展示某些无法由Godbillon-Vey类和Fuks-Lodder-Kotschick类描述的类，以及特征类的非平凡变形。

Abelian gerbes, generalized geometries and foliations of small exotic R^4

In the paper we prove the existence of the strict but relative relation between small exotic $\mathbb{R}^{4}$ for a fixed radial family of DeMichelis-Freedman type, and cobordism classes of codimension one foliations of $S^{3}$ distinguished by the Godbillon-Vey invariant, $GV\in H^{3}(S^{3},\mathbb{R})$ (represented by a 3-form). This invariant can be integrated to get the Godbillon-Vey number. For a fixed radial family, we will show that the isotopy classes (invariance w.r.t. small diffeomorphisms or coordinate transformations) of all members in this family are distinguished by the Godbillon…

Transversely holomorphic flows and contact circles on spherical 3-manifolds

Motivated by the moduli theory of taut contact circles on spherical 3-manifolds, we relate taut contact circles to transversely holomorphic flows. We give an elementary survey of such 1-dimensional foliations from a topological viewpoint. We describe a complex analogue of the classical Godbillon-Vey invariant, the so-called Bott invariant, and a logarithmic monodromy of closed leaves. The Bott invariant allows us to formulate a generalised Gauß-Bonnet theorem. We compute these invariants for the Poincaré foliations on the 3-sphere and derive rigidity statements, including a uniformisation theo…