关于一类自融合三维流形的Heegaard亏格

On a class of Heegaard genuses of self-amalgamated 3-manifolds

主要研究一个三维流形沿着自身的两个环面分支粘合后所得的三维流形在(关于原流形的Heegaard距离的)一定条件下的Heegaard亏格的非退化问题.设M是一个紧致连通定向的3-流形,T1,T2是M的边界上的两个环面分支,h:T1→T2为一个反向同胚,M′是M通过h粘合T1和T2所得到的定向3-流形.笔者证明了如下结果:如果M有一个Heegaard分解V∪SW,满足T1,T2_V或_W,且D(S)≥2g(M,T1∪T2)+1,则有g(M′)=g(M,T1∪T2)+1.

Let M be a compact connected oriented 3-manifold, T1, T2 two torus components of 8M, h： T1→T2 a orientation-reversing homeomorphism,M＇ is the manifold obtained from M by attaching T1 and T2 via h. The main result of the present paper is as follows： If M has a Heegaard splitting V∪sW such that T1 ,T2 belong to δ_V or δ_W,and D（S）≥2g（M, T1∪ T2）＋1, then g（M＇）=g（M, T1 ∪ T2） ＋1.