On amalgamations of Heegaard splittings with high distance

Abstract

Let M be a compact, orientable 3-manifold and F an essential closed surface which cuts M into M 1 and M 2 . Suppose that M i has a Heegaard splitting, and the amalgamation of V 1 ∪ S 1 W 1 and V 2 ∪ S 2 W 2 is the unique minimal Heegaard splitting of M up to isotopy.

Amalgamations of Heegaard splittings in 3–manifolds without some essential surfaces

“…to obtain subsurfaces Q In the following, we give an application of Theorem 4.2. Remark The condition in Theorem 4.2 is weaker than that in the main results of Kobayashi and Qiu [6] and Yang and Lei [19].…”

“…On the other hand, it has been shown that under some conditions on the manifolds, or the gluing maps, the equality g.M / D g.M 1 / C g.M 2 / g.F / holds; see Kobayashi and Qiu [6], Lackenby [8], Lei and Yang [9], Li [10], Souto [18], Yang and Lei [19] and others.…”

A sufficient condition for the genus of an amalgamated 3-manifold not to go down

“…Lemma 2.3 [15]. Let M = V ∪ S W be a strongly irreducible Heegaard splitting and F a 2-sided essential surface (not a disk or 2-sphere) in M .…”

“…In the papers of Kobayashi-Qiu [4] and Yang-Lei [15], they described a sufficient condition for the equality to hold in term of "high" distance of Heegaard splittings of the submanifolds, see, for example, Theorem 1.3 [15].…”

A Lower Bound for the Genus of Self-Amalgamation of Heegaard Splittings

“…On the other hand, it has been shown that under some conditions on the manifolds, the gluing maps, or the distances of the factor manifolds, the equality g(M ) = g(M 1 ) + g(M 2 ) − g(F ) holds, see [9], [10], [20], [7] and [21] etc.…”