Leaves Without Holonomy

Abstract

We prove the following result. THEOREM. Let M be a paracompact manifold with a foliation of codimension k. Let T be the union of all leaves with trivial holonomy. Then T is a dense G8 in M. This result is also due independently to G. Hector [3], who has shown how useful it can be in understanding the geometry of certain foliated manifolds. In such applications one sometimes needs a form of this theorem which applies to foliated subspaces, for example a minimal subset of a foliation. In fact our proof goes through unaltered in the situation where M is a locally compact, paracompact, Hausdorff foliated space such that each plaque is locally connected. We do not need to assume that M is a manifold. (We recall that locally compact Hausdorff spaces satisfy the Baire category theorem.) Our treatment of the result differs from that of Hector in several respects. Firstly we give complete details of the proof. Secondly we allow the manifold which is foliated to be non-compact. Thirdly we do not restrict the differentiability class of the foliation. Later we will give an example to show that T may be empty if M is not paracompact.