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Local Properties of the Embedding of a Graph in a Three-Manifold

Published online by Cambridge University Press:  20 November 2018

D. R. McMillan Jr.
D. R. McMillan Jr.*
Affiliation:
The University of Virginia
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Let G be a finite graph topologically embedded in the interior of a 3-manifold M. Doyle (4) and Debrunner and Fox (3) have noted that the following local homotopy condition at each point p ∈ G is necessary in order for the embedding of G to be tame:

For each sufficiently small open set U containing p, there is an open set V such that p ∈ V ⊂ U and if W is any connected open set such that p ∈ W ⊂ V, then the image under the inclusion homomorphism i*: π1(W — G) → π1(U — G) is a free group on n — 1 generators.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 517 - 528
Copyright
Copyright © Canadian Mathematical Society 1966

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