Handlebody splittings of 3-manifolds with bounday

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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.