3-流形相对亏格的可加性(英文)

The additivity of the relative genus of 3-manifolds

设M为一个紧致连通可定向的3-流形,M=F1∪F2为M的分支的无交并.M的一个Hee-gaard分解V1∪SV2称为是一个相对于(M;F1,F2)的Heegaard分解,若-V1=F1且-V2=F2.用g(M;F1,F2)来表示M的相对于(M;F1,F2)的所有Heegaard分解中的最小亏格,称之为M的相对于(M;F1,F2)的Heegaard亏格(或简称为M的相对亏格)。证明了3-流形的相对亏格在连通和下是可加的。

Let M be a compact connected orientable 3 - manifold, and OM = F1 U F2 a disjoint union of components of aM. A Heegaard splitting V1 UsV2 of M is called a relative Heegaard splitting for （ M; F1, F2 ） if a _ V1 = F1 and a _ V2 = F2. By g （ M; F1, F2 ） denote the relative Heegaard genus of M, which is equal to the minimal genus of all relative Heegaard splittings for （ M; F1, F2 ）. The obtained main result is that the relative genus of 3 -manifolds are additive under connected sum.