Foliations by planes and Lie group actions

J. A. Álvarez López¹ ; J. L. Arraut² ; C. Biasi³

¹ Departamento de Xeometría e Topoloxía Facultade de Matemáticas Universidade de Santiago de Compostela 15706 Santiago de Compostela, Spain
² Departamento de Matemática nstituto de Matemática e Computação Universidade de São Paulo Campus de São Carlos Caixa Postal 668 13560-970 São Carlos SP, Brasil
³ Departamento de Matemática Instituto de Matemática e Computação Universidade de São Paulo Campus de São Carlos Caixa Postal 668 13560-970 São Carlos SP, Brasil

Annales Polonici Mathematici, Tome 82 (2003) no. 1, pp. 61-69.

Résumé

Let $N$ be a closed orientable $n$-manifold, $n\ge 3$, and $K$ a compact non-empty subset. We prove that the existence of a transversally orientable codimension one foliation on $N\setminus K$ with leaves homeomorphic to ${{\mathbb R}}^{n-1}$, in the relative topology, implies that $K$ must be connected. If in addition one imposes some restrictions on the homology of $K$, then $N$ must be a homotopy sphere. Next we consider $C^{2}$ actions of a Lie group diffeomorphic to ${\mathbb R}^{n-1}$ on $N$ and obtain our main result: if $K$, the set of singular points of the action, is a finite non-empty subset, then $K$ contains only one point and $N$ is homeomorphic to $S^{n}$.

Détail

DOI : 10.4064/ap82-1-7

Keywords: closed orientable n manifold compact non empty subset prove existence transversally orientable codimension foliation setminus leaves homeomorphic mathbb n relative topology implies connected addition imposes restrictions homology homotopy sphere consider actions lie group diffeomorphic mathbb n obtain main result set singular points action finite non empty subset contains only point homeomorphic