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

Abstract

Let M be a compact connected 3-submanifold of the 3-sphere S ³ with one boundary component F such that there exists a collection of n pairwise disjoint connected orientable surfaces S = {S ₁, · · ·, S _n} properly embedded in M, ∂S = {∂S ₁, · · ·, ∂S _n} 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 _n.

Keywords

Complete surface system / Complete spanning curve system / Heegaard diagram / Handlebody addition

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