Homology-genericity, horizontal Dehn surgeries and ubiquity of rational homology 3-spheres
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- by Jiming Ma
- Proc. Amer. Math. Soc. 140 (2012), 4027-4034
- DOI: https://doi.org/10.1090/S0002-9939-2012-11224-9
- Published electronically: March 7, 2012
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Abstract:
In this paper, we show that rational homology 3-spheres are ubiquitous from the viewpoint of Heegaard splitting. Let $M=H_{+}\cup _{F} H_{-}$ be a genus $g$ Heegaard splitting of a closed $3$-manifold and $c$ be a simple closed curve in $F$. Then there is a 3-manifold $M_{c}$ which is obtained from $M$ by horizontal Dehn surgery along $c$. We show that for $c$ such that the homology class $[c]$ is generic in the set of curve-represented homology classes $\mathscr {H}_{s} \subset H_{1}(F)$, $rank(H_{1}(M_{c},\mathbb {Q}))<max \{1,rank(H_{1}(M,\mathbb {Q})$}. As a corollary, for a set of curves $\{c_1,c_2,\ldots , c_{m}\}$, $m \geq g$, such that each $[c_{i}]$ is generic in $\mathscr {H}_{s} \subset H_{1}(F)$, $M_{(c_1,c_2,\ldots , c_{m})}$ is a rational homology 3-sphere.References
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Bibliographic Information
- Jiming Ma
- Affiliation: School of Mathematical Sciences, Fudan University, Shanghai, People’s Republic of China 200433
- Email: majiming@fudan.edu.cn
- Received by editor(s): January 31, 2010
- Received by editor(s) in revised form: June 3, 2010, September 17, 2010, March 4, 2011, and April 26, 2011
- Published electronically: March 7, 2012
- Additional Notes: The author was supported in part by RFDP 200802461001 and NSFC 10901038.
- Communicated by: Daniel Ruberman
- © Copyright 2012 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 140 (2012), 4027-4034
- MSC (2010): Primary 57M27, 57M99
- DOI: https://doi.org/10.1090/S0002-9939-2012-11224-9
- MathSciNet review: 2944742