Geodesic
Taming Wild Simple Closed Curves with Monotone Maps
Canadian journal of mathematics, Tome 24 (1972) no. 5, pp. 768-788

Voir la notice de l'article provenant de la source Cambridge University Press

Hempel [6, Theorem 2] proved that if S is a tame 2-sphere in E3 and f is a map of E3 onto itself such that f|S is a homeomorphism and f(E3 - S) = E 3- f(S), then f(S) is tame. Boyd [4] has shown that the converse is false; in fact, if S is any 2-sphere in E 3, then there is a monotone map f of E 3 onto itself such that f |S is a homeomorphism, f(E3 — S) = E 3 — f(S), and f(S) is tame.It is the purpose of this paper to prove that the corresponding converse for simple closed curves in E3 is also false. We show in Theorem 4 that if J is any simple closed curve in a closed orientable 3-manifold M3 , then there is a monotone map f : M 3 → S3 such that f |J is a homeomorphism, f(J) is tame and unknotted, and f(M3 - J) = S3 - f(J).In Theorem 1 of § 2, we construct a cube-with-handles neighbourhood of a simple closed curve in an orientable 3-manifold. 
Boyd, W. S.; Wright, A. H. Taming Wild Simple Closed Curves with Monotone Maps. Canadian journal of mathematics, Tome 24 (1972) no. 5, pp. 768-788. doi: 10.4153/CJM-1972-074-4
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