Topological Symmetry Groups and Mapping Class Groups for Spatial Graphs

摘要

By a graph we shall mean the underlying space of a finite connected simplicial complex of dimension 1. A spatial graph is a graph embedded in a 3-manifold. The theory of spatial graphs is a generalization of classical knot theory. For a spatial graph Γ in S~3, the mapping class group MCG(S~3, Γ) (resp., MCG+(S~3,Γ)) is denned as the group of isotopy classes of the self-homeomorphisms (resp., orientation-preserving self-homeomorphisms) of S~3 that preserve Γ setwise. The cardinality of the group describes how many symmetries the spatial graph admits. In [16] it is shown that the group MCG(S~3, Γ) is always finitely presented.