On 3-submanifolds of S which admit complete spanning curve systems.

Let M be a compact connected 3-submanifold of the 3-sphere S with one boundary component F such that there exists a collection of n pairwise disjoint connected orientable surfaces S = { S , · · ·, S } properly embedded in M, ∂ S = {∂ S , · · ·, ∂ S } is a complete curve system on F. We call S a complete surface system for M, and ∂ S a complete spanning curve system for M. In the present paper, the authors show that the equivalent classes of complete spanning curve systems for M are unique, that is, any complete spanning curve system for M is equivalent to ∂ S. As an application of the result, it is shown that the image of the natural homomorphism from the mapping class group M( M) to M( F) is a subgroup of the handlebody subgroup H . [ABSTRACT FROM AUTHOR]