Pushing arcs and graphs around in handlebodies. (English) Zbl 0868.57024

Johannson, Klaus (ed.), Low-dimensional topology. Proceedings of a conference, held May 18-26, 1992 at the University of Tennessee, Knoxville, TN, USA. Cambridge, MA: International Press. Conf. Proc. Lect. Notes Geom. Topol. 3, 163-171 (1994).
An oriented 3-manifold obtained by attaching a finite collection of 1-handles to the 3-ball is called a handlebody. This work deals with the problem of moving arcs and graphs in handlebodies and planarity of graphs.
Particularly graphs in 3-space and arc families in \(F\times I\) where \(F\) is a surface, are discussed. This is a continuation of the work of Y.-Q. Wu on planar graphs in 3-manifolds and of N. Robertson, P. D. Seymour and R. Thomas on linkless embeddings of graphs in 3-space. In analogy to the families of arcs in \(F\times I\), a graph \(\Gamma\) in \(S^3\) in taken and called cycle-trivial if every imbedded cycle in it bounds a disc having disjoint interior from \(\Gamma\). Then it is proven that if \(\Gamma\) is finite and is the disjoint union of two subgraphs \(B\) and \(C\), then the necessary and sufficient conditions for it to be made cycle-trivial over \(C\) is \(S^3-\eta (B' \cup C)\) is a connected sum of handlebodies for every \(B' \subset B\). It is also proven that this is still true when \(\Gamma\) is a graph in a connected sum of handlebodies.
For the entire collection see [Zbl 0816.00026].


57N35 Embeddings and immersions in topological manifolds
57M99 General low-dimensional topology