Foliated and compactly supported isotopies of tubular neighborhoods

Abstract

Let $\mathcal{F}$ be a foliation with a "singular" submanifold $\Sigma$ on a smooth manifold $A$ and $p:E\to \Sigma$ be a tubular neighborhood of $\Sigma$ in $A$. Under certain "scalability" assumptions on $\mathcal{F}$ near $\Sigma$, we prove that every leaf preserving diffeomorphism $h$ of $A$ is isotopic via a leaf preserving isotopy to a diffeomorphism which coincides with some vector bundle morphism of $E$ near $\Sigma$. This result is simultaneously a foliated, compactly supported and parametrized 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 at $0$. Particular cases of such a "linearization theorem" allowed recently to compute homotopy types of leaf preserving diffeomorphism groups of polar Morse-Bott foliations on lens spaces and the twisted bundle $S^2\tilde{\times }{S}^1$. In general, that "linearization theorem" can be used to reduce the study of the homotopy type of the group of leaf preserving diffeomorphisms of $\mathcal{F}$ to an investigation of the following three groups (being independent in a certain sense): (a) diffeomorphism group of $\Sigma$, (b) the group of leaf preserving vector bundle isomorphisms of $E$, and (c) the group of leaf preserving diffeomorphisms of $A$ supported in $A\setminus \Sigma$, i.e. out of the "singularity" $\Sigma$.

arXiv comment37 pages, 2 figures. Essentially reworked

Cited by in corpus (1)

• Diffeomorphism groups of Morse-Bott foliation on the solid Klein bottle by Klein bottles parallel to the boundary