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Summary: PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 1 15, Number 4, AUGUST 1992
HANDLEBODY COMPLEMENTS IN THE 3-SPHERE:
A REMARK ON A THEOREM OF FOX
MARTIN SCHARLEMANN
(Communicated by James E. West)
ABSTRACT.Let W be a compact 3-dimensional submanifold of S3,and C
be a collection of disjoint simple closed curves on 6' W. We give necessary and
sufficient conditions (one extrinsic, one intrinsic) for W to have an imbedding
in S3 SO that S3- W is a union of handlebodies, and C contains a complete
collection of meridia for these handlebodies.
A theorem of Fox [F] says that any connected compact 3-dimensional sub-
manifold W of S3 is homeomorphic to the complement of a union of han-
dlebodies in S 3 . Call a collection C of simple closed curves in d W proto-
meridinal if there is some embedding of W in S3 SO that the complement of
W is a union of handlebodies and C contains a complete collection of meridia
for this handlebody. It is natural to ask whether a given collection C of simple
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