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A necessary and sufficient condition for a surface sum of two handlebodies to be a handlebody

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Abstract

The main results of the paper are that we give a necessary and sufficient condition for a surface sum of two handlebodies along a connected surface to be a handlebody as follows: (1) The annulus sum H = H1∪AH2 of two handlebodies H1 and H2 is a handlebody if and only if the core curve of A is a longitude for either H1 or H2; (2) Let H = H1∪Sg,b H2 be a surface sum of two handlebodies H1 and H2 along a connected surface S = Sg,b,,b ≥ 1, ni = g(Hi) ≥ 2, i = 1, 2. Suppose that S is incompressible in both H1 and H2. Then H is a handlebody if and only if there exists a basis J = {J}1,…, Jm with a partition (J1, J2) of J such that J1 is primitive in H1 and J2 is primitive in H2.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11431009 and 11671064), the Fundamental Research Funds for the Central Universities (Grant No. DUT19LK15) and Ministry of Science and Education of Russia (Grant No. 1.13557.2019/13.1). The authors thank the referees most warmly for their many helpful suggestions to revise the paper.

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Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

    Fengchun Lei, He Liu & Fengling Li

  2. Regional Scientific and Educational Mathematical Center, Department of Mechanics and Mathematics, Tomsk State University, Tomsk, 634050, Russia

    Andrei Vesnin

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  1. Fengchun Lei
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  2. He Liu
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  3. Fengling Li
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  4. Andrei Vesnin
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Correspondence to Fengling Li.

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Lei, F., Liu, H., Li, F. et al. A necessary and sufficient condition for a surface sum of two handlebodies to be a handlebody. Sci. China Math. 63, 1997–2004 (2020). https://doi.org/10.1007/s11425-019-1647-9

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  • DOI: https://doi.org/10.1007/s11425-019-1647-9

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