Foliations by planes and Lie group actions
Volume 82 / 2003
Annales Polonici Mathematici 82 (2003), 61-69
MSC: Primary 57S25; Secondary 57R25, 57R30.
DOI: 10.4064/ap82-1-7
Abstract
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 $S^{n}$.
