A necessary and sufficient condition for a $3$-manifold to have Heegaard genus one
HTML articles powered by AMS MathViewer
- by Joel Hass and Abigail Thompson PDF
- Proc. Amer. Math. Soc. 107 (1989), 1107-1110 Request permission
Abstract:
Let $M$ be a closed $3$-manifold. R. H. Bing showed that $M$ is homeomorphic to ${S^3}$ if and only if every simple closed curve in $M$ can be isotoped to lie inside a $3$-ball. We generalize this to show that there is a solid torus $T$ imbedded in $M$ such that every simple closed curve in $M$ can be isotoped to lie in $T$ if and only if $M$ has a genus one Heegaard splitting.References
- R. H. Bing, Necessary and sufficient conditions that a $3$-manifold be $S^{3}$, Ann. of Math. (2) 68 (1958), 17–37. MR 95471, DOI 10.2307/1970041
- C. McA. Gordon and José María Montesinos, Fibred knots and disks with clasps, Math. Ann. 275 (1986), no. 3, 405–408. MR 858286, DOI 10.1007/BF01458613
- 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
- William Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics, vol. 43, American Mathematical Society, Providence, R.I., 1980. MR 565450
- D. R. McMillan Jr., On homologically trivial $3$-manifolds, Trans. Amer. Math. Soc. 98 (1961), 350–367. MR 120639, DOI 10.1090/S0002-9947-1961-0120639-0
- Robert Myers, Open book decompositions of $3$-manifolds, Proc. Amer. Math. Soc. 72 (1978), no. 2, 397–402. MR 507346, DOI 10.1090/S0002-9939-1978-0507346-5
- Robert Myers, Simple knots in compact, orientable $3$-manifolds, Trans. Amer. Math. Soc. 273 (1982), no. 1, 75–91. MR 664030, DOI 10.1090/S0002-9947-1982-0664030-0
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 107 (1989), 1107-1110
- MSC: Primary 57N10
- DOI: https://doi.org/10.1090/S0002-9939-1989-0984792-4
- MathSciNet review: 984792