HOMOLOGY-GENERICITY, HORIZONTAL DEHN SURGERIES AND UBIQUITY OF RATIONAL HOMOLOGY 3-SPHERES.

In this paper, we show that rational homology 3-spheres are ubiquitous from the viewpoint of Heegaard splitting. Let M = H+ UF H_ be a genus g Heegaa;rd splitting of a closed 3-manifold and c be a simple closed curve in F. Then there is a 3-manifold Mc 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 Hs ⊂ H1(F), rank(H1(Mc,Q)) < max{ 1, rank(H1(M,Q)}. As a corollary, for a set of cusub>, c2, hellip;, cm}, m ≥ g, such that each [ci] is generic in Hs ⊂ H1(F), M(c1,c2,…,cm) is a rational homology 3-sphere. [ABSTRACT FROM AUTHOR]