Foliated and compactly supported isotopies of regular neighborhoods

Abstract

Let $\mathcal{F}$ be a foliation with a "singular" submanifold $B$ on a smooth manifold $M$ and $p:E \to B$ be a regular neighborhood of $B$ in $M$. Under certain "homogeneity" assumptions on $\mathcal{F}$ near $B$ we prove that every leaf preserving diffeomorphism $h$ of $M$ is isotopic via a leaf preserving isotopy to a diffeomorphism which coincides with some vector bundle morphism of $E$ near $B$. This result is mutually a foliated and compactly supported variant of a well known statement that every diffeomorphism $h$ of $\mathbb{R}^n$ fixing the origin is isotopic to the linear isomorphism induced by its Jacobi matrix of $h$ at $0$. We also present applications to the computations of the homotopy type of the group of leaf preserving diffeomorphisms of $\mathcal{F}$.