On the union stabilization of two Heegaard splittings.pdf

a r X i v : 0 8 0 8 .0 5 4 7 v 1 [ m a t h .G T ] 5 A u g 2 0 0 8 ON THE UNION STABILIZATION OF TWO HEEGAARD SPLITTINGS JUNG HOON LEE Abstract. Let two Heegaard splittings V1 ∪ W1 and V2 ∪ W2 of a 3- manifold M be given. We consider the union stabilization M = V ∪ W which is a common stabilization of V1 ∪ W1 and V2 ∪ W2 having the property that V = V1 ∪ V2. We show that any two Heegaard splittings of a 3-manifold have a union stabilization. We also give some examples with numerical bounds on the minimal genus of union stabilization. On the other hand, we give an example of a candidate for which the minimal genus of union stabilization is strictly larger than the usual stable genus — the minimal genus of common stabilization. 1. Introduction A Heegaard splitting M = H1∪S H2 of a 3-manifold M is a decomposition of M into two handlebodies H1 and H2 and it is well known that every compact 3-manifold admits Heegaard splittings. A stabilization of a Heegaard splitting H1 ∪S H2 is an operation that results in a new Heegaard splitting H ′ 1 ∪S′ H ′ 2 , with the genus increased by one. Here H ′ 1 is obtained by adding trivial 1-handle to H1 whose core is ?-parallel in H2 and H ′ 2 is obtained by removing the 1-handle from H2. When two Heegaard splittings V1 ∪ W1 and V2 ∪ W2 of a 3-manifold are given, they become isotopic after a sequence of stabilizations [8]. The mini- mal genus among the common stabilizations is called the stable genus. Now we introduce a new concept slightly stronger than the common stabilization. A Heegaard splitting V ∪W is a union stabilization of V1∪W1 and V2∪W2 if ? V ∪ W is a common stabilization of V1 ∪ W1 and V2 ∪ W2. ? V = V1 ∪ V2 In section 2, we show that any two Heegaard splittings have a union stabilization, similar to the results of [8]. The minimal genus among the union stabilizations is called the union genus. Remark 1.1. (1) From the definition of the union stabilization, it follows that W = W1 ∩ W2. 1991 Mathematics Subject Class