The conjugacy problem for boundary loops in $3$-manifolds
HTML articles powered by AMS MathViewer
- by Benny D. Evans PDF
- Trans. Amer. Math. Soc. 240 (1978), 53-64 Request permission
Abstract:
A geometric solution of the word problem for fundamental groups of compact, orientable, irreducible, sufficiently large 3-manifolds has been given by F. Waldhausen. We present here a solution of a restricted version of the conjugacy problem for this same class of 3-manifolds; however, the conjugacy problem for 3-manifolds remains in general unsolved. The main results is that there is an algorithm that will determine for any two loops ${L_1},{L_2}$ in the boundary of a compact, orientable, irreducible sufficiently large 3-manifold M if ${L_1}$, is freely homotopic in M to ${L_2}$.References
- M. Dehn, Über unendliche diskontinuierliche Gruppen, Math. Ann. 71 (1911), no. 1, 116–144 (German). MR 1511645, DOI 10.1007/BF01456932 C. F. Feustel, The torus theorem and its applications (preprint).
- Wolfgang Haken, Some results on surfaces in $3$-manifolds, Studies in Modern Topology, Math. Assoc. America, Buffalo, N.Y.; distributed by Prentice-Hall, Englewood Cliffs, N.J., 1968, pp. 39–98. MR 0224071
- Wolfgang Haken, Über das Homöomorphieproblem der 3-Mannigfaltigkeiten. I, Math. Z. 80 (1962), 89–120 (German). MR 160196, DOI 10.1007/BF01162369
- William Jaco, Roots, relations and centralizers in three-manifold groups, Geometric topology (Proc. Conf., Park City, Utah, 1974) Lecture Notes in Math., Vol. 438, Springer, Berlin, 1975, pp. 283–309. MR 0394668
- Friedhelm Waldhausen, The word problem in fundamental groups of sufficiently large irreducible $3$-manifolds, Ann. of Math. (2) 88 (1968), 272–280. MR 240822, DOI 10.2307/1970574 —, On the determination of some 3-manifolds by their fundamental groups alone, Proc. Internat. Sympos. on Topology and Its Applications (Herceg-Novi, August 1968, Yugoslavia), Savez Drušstava Mat., Fix. i Astronom., Belgrade, 1969, pp. 331-332. MR 42 #2416.
- Friedhelm Waldhausen, On irreducible $3$-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56–88. MR 224099, DOI 10.2307/1970594
Additional Information
- © Copyright 1978 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 240 (1978), 53-64
- MSC: Primary 55A05; Secondary 57A10
- DOI: https://doi.org/10.1090/S0002-9947-1978-0478129-4
- MathSciNet review: 0478129