On the First Homology of the Groups of Foliation Preserving Diffeomorphisms for Foliations with Singularities of Morse Type

Abstract

Let ${\mathbf{R}}^n$ be an $n$-dimensional Euclidean space and ${\mathcal{F}}_{\phi }$ be the foliation defined by levels of a Morse function $\phi :{\mathbf{R}}^n\to \mathbf{R}$. We determine the first homology of the identity component of the foliation preserving diffeomorphism group of $({\mathbf{R}}^n,{\mathcal{F}}_{\phi })$. Then we can apply it to the calculation of the first homology of the foliation preserving diffeomorphism groups for codimension one compact foliations with singularities of Morse type.