Abstract
Let M be a compact connected 3-submanifold of the 3-sphere S3 with one boundary component F such that there exists a collection of n pairwise disjoint connected orientable surfaces S = {S1, · · ·, S n } properly embedded in M, ∂S = {∂S1, · · ·, ∂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 .
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Acknowledgments
The authors wish to thank the referee most warmly for numerous suggestions that have improved the exposition of this paper.
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This work was supported by the National Natural Science Foundation of China (Nos. 11329101, 11431009, 11329101, 11471151, 11401069) and the grant of the Fundamental Research Funds for the Central Universities (No.DUT14LK12).
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Zhao, Y., Lei, F. & Li, F. On 3-submanifolds of S3 which admit complete spanning curve systems. Chin. Ann. Math. Ser. B 38, 1373–1380 (2017). https://doi.org/10.1007/s11401-017-1045-1
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DOI: https://doi.org/10.1007/s11401-017-1045-1