Stability of foliations

Abstract:

Let X be a compact manifold and let k be an integer. It is shown that the set of homeomorphism conjugacy classes of germs at X of foliations of codimension k and the set of homeomorphism conjugacy classes of (holonomy) representations of ${\prod _1}(X)$ in the group of germs at 0 of 0-fixed self-diffeomorphisms of ${{\text {R}}^k}$ are homeomorphic when given appropriate topologies. Stable foliation germs and stable holonomy representations correspond under this homeomorphism. It is shown that there are no stable foliation germs at a toral leaf if the dimension of the torus is greater than one.