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LEAVES WITHOUT HOLONOMY D. B. A. EPSTEIN, K. C. MILLETT AND D. TISCHLER
 

Summary: LEAVES WITHOUT HOLONOMY
D. B. A. EPSTEIN, K. C. MILLETT AND D. TISCHLER
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 para-
compact. We note that if M is a paracompact manifold, then the interior of T may
e empty, and we will give an example which displays this behaviour.
Proof of the theorem. Let / be an indexing set for a family of admissible charts

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics