关于非强不可约的Heegaard分解

On Non-Strongly Irreducible Heegaard Splittings

用δ(M,F)表示亏格为n的可定向闭3-流形M的Heegaard分解(M,F)的圆片指数,本文证得:(1)(M,F)不是强不可约的当且仅当δ(M,F)≥2;(2)(M,F)是可约的当且仅当δ(M,F)≥max(2,n};(3)若2≤δ(M,F)<n,则M包含一个正亏格的不可压缩曲面。

For a Heegaard splitting (M,F) with genus n of an orientable closed 3-manifold M, we introduce the concept of disk index, δ(M,F), of (M,F), and infer that(1) (M,F) is not strongly irreducible iff δ(M,F)≥2;(2) (M,F) is reducible iff δ(M,F)≥max{2,n};(3) 2≤δ(M,F)<n implies that M contains a 2-sided incompressible surface with positive genus.Thus we supply and improve the main work of Casson and Gordon in [3].