Heegaard Genera of Annular 3-Manifolds

Abstract

Let M be a compact orientable 3-manifold, and A an essential annulus which cuts M into two 3-manifolds M1 and M2. We denote by g(M) the Heegaard genus of M. In this paper, we will give a lower bound to g(M) when Mi contains no bounded essential surfaces with large Euler's characteristic, where all boundary components of the essential surfaces lie in A.

Some Sufficient Conditions for Heegaard Genera to Be Additive Under Annulus Sum

“…And with the weaker assumptions, we obtain the stronger conclusion that the minimal Heegaard splitting of M is in some sense unique. Hence we remark that the situations here are quite different from those in [2], [8] and [17], and the arguments there are not applicable to the main cases here.…”

“…By arguments similar to those in the proof of Lemma 3.1, we know that S ∩ M 2 is bicompressible in M 2 while S ∩ (F 1 × I ) is ∂-parallel in F 1 × I . If all components of S ∩ (F 1 × I ) are parallel to the same component of F 1 1 and F 2 1 , say F 1 1 , then in V , a component of A ∩ V cuts off a trivial compression body F 1 × I from V , but this is impossible since the component of A ∩ V is a spanning annulus in V . Hence at least one component of S ∩ (F 1 × I ) is parallel to F 1 1 and at least one component of…”

“…Some sufficient conditions for tunnel number of knots not to go down under connected sum are given in [17]. When both A 1 and A 2 are non-separating on the corresponding boundary component, there are some sufficient conditions for the equality g(M) = g(M 1 ) + g(M 2 ) to hold, see [2], [8].…”

“…Note that papers [8] and [17] only consider the case that both A 1 and A 2 are non-separating. In [2], the bound of the Heegaard distance for the additivity to hold is "d(S i ) ≥ 2(g(M 1 ) + g(M 2 )), i = 1, 2", in other words, the bound of d(S 1 ) (d(S 2 ), resp.) of M 1 (M 2 , resp.)…”