Compressing handlebodies with holes
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 \(\partial M\).
scientific article; zbMATH DE number 4119252