Compressing handlebodies with holes. (English) Zbl 0683.57004

Menasco, W.; Thompson, A.

Topology 28, No. 4, 485-494 (1989).

A theorem of Fox asserts that a 3-manifold M with connected boundary can be embedded in 3-space if and only if it is possible to glue 2-handles along the boundary of M so that the resulting manifold is a handlebody, namely can be cut along a family of disjoint embedded disks to obtain a 3-ball. The authors introduce a notion of handle-wormhole structure on M hich keeps track of this process of adding 2-handles and cutting along discs. The main result of the article is that, if M has compressible boundary, any handle-wormhole structure for M can be modified by a sequence of relatively simple moves so as to avoid a compression disk for \(\partial M\).

Reviewer: F.Bonahon

MSC:

57N10	Topology of general \(3\)-manifolds (MSC2010)
57R65	Surgery and handlebodies
57M35	Dehn’s lemma, sphere theorem, loop theorem, asphericity (MSC2010)

Keywords:

surfaces in 3-space; hollow handlebodies; Heegaard splittings; 3- manifold; handle-wormhole structure; adding 2-handles; cutting along discs