Handlebody splittings of 3-manifolds with bounday

我们称一个具有非空边界的紧致连通可定向3-流形M为加边3-流形。一个有边的3-流形H称为亏格g的一个分裂体当且仅当H是g个实环面副本的圆盘和（=边界连通和）.众所周知，一个闭的（=紧的，无边界的）、连通的、可定向的3-流形M可分解为两个同胚的分裂体，这种分裂称为M的Heegaard分裂. Downig证明了每一个加边三维流形都可以分解为两个同胚的同构体，L。G. Roeling讨论了具有连通边界的加边三维流形的这些分解。本文的研究结果是：（1）将Downing的结果作了一些修改，（2）将Roeling的结果推广到具有多个边界的有边三维流形，（3）用Casson和Gordon的方法建立了这类分解的一个Haken型定理，（4）讨论了这类分解的另一个Haken型定理。

We call a compact, connected, orientable 3-manifold M with nonempty boundary ∂M a bordered 3-manifold. A bordered 3-manifold H is said to be a handlebody of genus g iff H is the disk-sum (= the boundary connected-sum) of g copies of the solid-torus.It is well-known that a closed (=compact, without boundary) , connected, orientable 3-manifold M is decomposed into two homeomorphic handlebodies ; and such a splitting is called a Heegaard splitting for M.On the other hand, in 1970 J. S. Downig proved that every bordered 3-manifold can be decomposed into two homeomorphic handlebodies, and L. G. Roeling discoursed on these decompositions for bordered 3-manifolds with connected boundary. The research results are (1) to report the Downing's results in slightly modified forms, (2) to generalize the Roeling's results to borderd 3-manifolds with several boundaries, (3) to formulate a Haken type theorem for these decompositions in the way of Casson and Gordon, and (4) to discuss another Haken type theorem for these decomposition with essential proper disks.