A necessary and sufficient condition for a -manifold to have Heegaard genus one

Authors:
Joel Hass and Abigail Thompson

Journal:
Proc. Amer. Math. Soc. **107** (1989), 1107-1110

MSC:
Primary 57N10

MathSciNet review:
984792

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Abstract: Let be a closed -manifold. R. H. Bing showed that is homeomorphic to if and only if every simple closed curve in can be isotoped to lie inside a -ball. We generalize this to show that there is a solid torus imbedded in such that every simple closed curve in can be isotoped to lie in if and only if has a genus one Heegaard splitting.

**[1]**R. H. Bing,*Necessary and sufficient conditions that a 3-manifold be 𝑆³*, Ann. of Math. (2)**68**(1958), 17–37. MR**0095471****[2]**C. McA. Gordon and José María Montesinos,*Fibred knots and disks with clasps*, Math. Ann.**275**(1986), no. 3, 405–408. MR**858286**, 10.1007/BF01458613**[3]**Wolfgang Haken,*Some results on surfaces in 3-manifolds*, Studies in Modern Topology, Math. Assoc. Amer. (distributed by Prentice-Hall, Englewood Cliffs, N.J.), 1968, pp. 39–98. MR**0224071****[4]**William Jaco,*Lectures on three-manifold topology*, CBMS Regional Conference Series in Mathematics, vol. 43, American Mathematical Society, Providence, R.I., 1980. MR**565450****[5]**D. R. McMillan Jr.,*On homologically trivial 3-manifolds*, Trans. Amer. Math. Soc.**98**(1961), 350–367. MR**0120639**, 10.1090/S0002-9947-1961-0120639-0**[6]**Robert Myers,*Open book decompositions of 3-manifolds*, Proc. Amer. Math. Soc.**72**(1978), no. 2, 397–402. MR**507346**, 10.1090/S0002-9939-1978-0507346-5**[7]**Robert Myers,*Simple knots in compact, orientable 3-manifolds*, Trans. Amer. Math. Soc.**273**(1982), no. 1, 75–91. MR**664030**, 10.1090/S0002-9947-1982-0664030-0

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DOI:
https://doi.org/10.1090/S0002-9939-1989-0984792-4

Article copyright:
© Copyright 1989
American Mathematical Society