Constructing isotopes on noncompact 3-manifolds

Brown, Marianne S.

doi:10.1090/S0002-9947-1973-0331393-X

Constructing isotopes on noncompact $3$-manifolds
HTML articles powered by AMS MathViewer

by Marianne S. Brown PDF

Trans. Amer. Math. Soc. 180 (1973), 237-263 Request permission

Abstract:

We consider the question “When are two homeomorphisms of a noncompact $3$-manifold onto itself isotopic?” Roughly, the answer is when they are homotopic to each othet. More precisely, this paper deals with the question for irreducible $3$-manifolds which either have an infinite hierarchy or have such a hierarchy after the removal of a compact set. Manifolds having the first property are called end-irreducible; the others are called eventually endirreducible. There are two results fot each type of manifold depending on whether the homotopy between the two homeomorphisms sends the boundary of the manifold into itself or not.

References

Kurventypen auf Flächen

156

Isotopie von Kurven auf orientbaren, geschossenen Flächen und ihr Ausammenhang mit der topologischen Deformation der Flächen

159

R. H. Bing, Locally tame sets are tame, Ann. of Math. (2) 59 (1954), 145–158. MR 61377, DOI 10.2307/1969836

On proper homotopy type

E. M. Brown and T. W. Tucker, On proper homotopy theory for noncompact $3$-manifolds, Trans. Amer. Math. Soc. 188 (1974), 105–126. MR 334225, DOI 10.1090/S0002-9947-1974-0334225-X
D. B. A. Epstein, Curves on $2$-manifolds and isotopies, Acta Math. 115 (1966), 83–107. MR 214087, DOI 10.1007/BF02392203

Isotopic unknotting in

Gordon M. Fisher, On the group of all homeomorphisms of a manifold, Trans. Amer. Math. Soc. 97 (1960), 193–212. MR 117712, DOI 10.1090/S0002-9947-1960-0117712-9
Wolfgang Haken, Über das Homöomorphieproblem der 3-Mannigfaltigkeiten. I, Math. Z. 80 (1962), 89–120 (German). MR 160196, DOI 10.1007/BF01162369
J. M. Kister, Isotopies in $3$-manifolds, Trans. Amer. Math. Soc. 97 (1960), 213–224. MR 120628, DOI 10.1090/S0002-9947-1960-0120628-5
D. R. McMillan Jr., Some contractible open $3$-manifolds, Trans. Amer. Math. Soc. 102 (1962), 373–382. MR 137105, DOI 10.1090/S0002-9947-1962-0137105-X
Edwin E. Moise, Affine structures in $3$-manifolds. VIII. Invariance of the knot-types; local tame imbedding, Ann. of Math. (2) 59 (1954), 159–170. MR 61822, DOI 10.2307/1969837
Friedhelm Waldhausen, On irreducible $3$-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56–88. MR 224099, DOI 10.2307/1970594

A certain open manifold whose group is unity

6

Simplicial spaces, nuclei, and $m$-groups

45

Arnold Shapiro and J. H. C. Whitehead, A proof and extension of Dehn’s lemma, Bull. Amer. Math. Soc. 64 (1958), 174–178. MR 103474, DOI 10.1090/S0002-9904-1958-10198-6

On the group of a certain linkage

Marianne Brown, Constructing isotopies in noncompact $3$-manifolds, Bull. Amer. Math. Soc. 78 (1972), 461–464. MR 298667, DOI 10.1090/S0002-9904-1972-12945-8