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

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.