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

摘要

In this paper, we show that rational homology 3-spheres are ubiquitous from the viewpoint of Heegaard splitting. Let M = H+ ∪ 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 H _s ? H _1(F), rank(H _1(M c,?)) < max{1, rank(H _1(M,?)}. As a corollary, for a set of curves {c _1, c _2,..., c _m}, m ≥ g, such that each [ci] is generic in H _s ? H _1(F), M(c _1,c _2,...,c _m) is a rational homology 3-sphere.