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Homology-genericity, horizontal Dehn surgeries and ubiquity of rational homology 3-spheres

Author: Jiming Ma
Journal: Proc. Amer. Math. Soc. 140 (2012), 4027-4034
MSC (2010): Primary 57M27, 57M99
Published electronically: March 7, 2012
MathSciNet review: 2944742
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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.

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Jiming Ma
Affiliation: School of Mathematical Sciences, Fudan University, Shanghai, People’s Republic of China 200433

Keywords: Rational homology 3-sphere, Heegaard splitting, homology-genericity.
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
Article copyright: © Copyright 2012 American Mathematical Society

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