Decomposing compact 3-manifolds into homeomorphic handlebodies
J. Scott Downing · Proceedings of the American Mathematical Society · 1970
In this paper all spaces and maps considered will be in the polyhedral category in the sense of Zeeman [4]. Thus all spaces can be triangulated and all maps and subspaces will be piecewise linear (PL). This will cause no restriction as every 3-manifold has a combinatorial triangulation [1]. Suppose M is a triangulated, closed 3-manifold. It is known [3, p. 219] that if N is a regular neighborhood of the 1-skeleton of the triangulation, then N and cl(M-N), the closure of its complement, are homeomorphic handlebodies. The first theorem of this paper gives a similar result for compact 3-manifolds with boundary.