Different Heegaard Structures On A 3-manifold

Since Heegaard spiliting play a key role in studying 3-manifold. Althoughsome namifold with di?enent unstabilized Heegaard spliting structue are give,but a Seifert manifold which has two di?erent unstabilized Heegaard spilitingwas ?rst given in this paper. And the example also show the complexity ofthe manifold which even has Heegaard spiliting of genus two. A compression body C is a 3-manifold obtained by adding 2-handles toF × I, where F is a connected closed surface, along a collection of pairwisedisjoint simple closed curves on F ×{0}, then capping o? any resulting 2-sphereboundary components with 3-balls. Denote by ?+C the surface F ×{1} in ?C,and ??C = ?C ??+C. When ??C = ?, C is a handlebody. When C = F ×I,C is a trivial compression body. Let M be a 3-manifold such that ?M has no 2-sphere components. AHeegaard splitting of M is a pair (V,W), where V,W are compression bodiessuch that V ∪W = M, and V ∩W = ?+V = ?+W = F. F is called a Heegaardsurface in M. The splitting is often denoted as V ∪F W, and the genus of Fis called the genus of the Heegaard splitting. A Heegaard splitting M = V ∪F W is said to be stabilized if there are twoproperly embedding disks D1 ? V and D2 ? W such that D1 intersects D2 inone point; otherwise, it is said to be unstabilized. It is easy to see that V ∪ Wis reducible if V ∪ W is stabilized. Let M be a compact 3-manifold, and S be an essential, separating closedsurface in M. We denote by M1,M2 the two components of M ? S. Now letMi = H1 ∪ H2 be a Heegaard splitting of Mi such that ??H2 = ??H1 = S. i i 1 2 第 1 页英文摘要Now M has a natural Heegaard splitting as follow: M = (??H1 × I) ∪ 1 ? handles in H1 and H1 1 1 2 ∪ 2 ? handles in H2 and H2 ∪ 3 ? handles . 1 2 In the other word, let H1 be the manifold obtained by attaching all 1-handels in H1 and H1 to ??H1 × I, then H1 is a compression body with 1 2 1??H1 = ??H1. Let H2 be the manifold obtained by attaching all 2-handles 1in H2 and H2 to ?+H1 × I then capping o? the possible spherical boundary 1 2components with 3-handles. Then M = H1 ∪H2 is a Heegaard splitting of M.Now we say H1 ∪ H2 is the amalgamation of H1 ∪ H2 and H1 ∪ H2. 1 1 2 2 Let M be a compact 3-manifold with boundary. If c is a simple closedcurves on ?M. We denote by τ(M,c) the manifold obtained by attaching a2-handle to M along regular neighborhood of c in ?M. Let H2 be a handlebody of genus two, and let D1, D2 be set of basis disksof H2. Set D3 be a separate disk of H2, such that D1 and D2 lie on oppositesides of D3, as shown in Figure 1. Suppose u1, v1, u2, v2 are four points on?D3 as shown in Figure 1. Suppose u1v1, u1v2, u2v1 be arcs on ?H2 as shown in Figure 1, then u1v1intersects D1 for four times while u1v2, u2v1 intersects D2 for just once. D1 D3 D2 u1 u2 v1 v2 第 2 页英文摘要 Figure 1 Since u1v1 meets D3 exactly in its two ends, we do tubing of D3 alongu1v1 to get proper surface D3(u1v1), that is, take a regular neighborhood ofu1v1 in ?H2 ? ?D3, and attach it to D3, then push it slightly so that weobtain a proper surface. Now u2v2 meets D3(u1v1) exactly in its two ends, wedo tubing of D3(u1v1) along u2v2 to get S = D3(u1v1)(u2v2), where the tubeN(u2v2) is thinner and closer to ?H2 as shown in Figure 2 Then S is a genusone surface with one boundary, and S is incompressible and separate in H2.The Proof can be ?...