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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].
##### MSC:
 57N35 Embeddings and immersions in topological manifolds 57M99 General low-dimensional topology
##### Keywords:
3-manifold; handlebody; graphs; embeddings