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Reducing and annular Dehn fillings

Author(s): Sangyop Lee
Journal: Trans. Amer. Math. Soc. 359 (2007), 227-247.
MSC (2000): Primary 57N10
Posted: August 15, 2006
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Abstract: If two Dehn fillings on a simple manifold create a reducible manifold and an annular manifold respectively, then the distance between those filling slopes is known to be at most two. Moreover, Eudave-Muñoz and Wu gave infinitely many examples of manifolds admitting reducing and annular Dehn fillings at distance two. In this paper, we complement their examples to establish a complete list of simple manifolds admitting such a pair of Dehn fillings.

References:

1.
M. Culler, C. McA. Gordon, J. Luecke, and P.B. Shalen, Dehn surgery on knots, Ann. of Math. 125 (1987), 237-300. MR 0881270 (88a:57026)

2.
J. Dean, Small Seifert-fibered Dehn surgery on hyperbolic knots, Algebr. Geom. Topol. 3 (2003), 435-472. MR 1997325 (2004m:57009)

3.
M. Eudave-Muñoz and Y.Q. Wu, Nonhyperbolic Dehn fillings on hyperbolic $ 3$-manifolds, Pacific J.Math. 190 (1999), 261-275. MR 1722893 (2000j:57035)

4.
C. McA. Gordon, Combinatorial Methods in Dehn surgery, Lectures at KNOTS '96 (Tokyo), Ser. Knots Everything, 15, World Sci. Publishing, River Edge, NJ, 1997, pp. 263-290. MR 1474525 (98m:57020)

5.
C. McA. Gordon, Boundary slopes on punctured tori in $ 3$-manifolds, Trans. Amer. Math. Soc. 350 (1998), 1713-1790. MR 1390037 (98h:57032)

6.
C. McA. Gordon, Small surfaces and Dehn fillngs, Geom. and Topol. Mono. 2 (1999), 177-199. MR 1734408 (2000j:57036)

7.
C. McA. Gordon and R. Litherland, Incompressible planar surfaces in $ 3$-manifolds, Topology Appl. 18 (1984), 121-144. MR 0769286 (86e:57013)

8.
C. McA. Gordon and J. Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989), 371-415. MR 0965210 (90a:57006a)

9.
C. McA. Gordon and J. Luecke, Dehn surgeries on knots creating essential tori. I, Comm. Anal. Geom. 3 (1995), 597-644. MR 1371211 (96k:57003)

10.
C. McA. Gordon and J. Luecke, Reducible manifolds and Dehn surgery, Topology 35 (1996), 385-409. MR 1380506 (97b:57013)

11.
C. McA. Gordon and J. Luecke, Toroidal and boundary-reducing Dehn fillings, Topology Appl. 93 (1999), 77-90. MR 1684214 (2000b:57030)

12.
C. McA. Gordon and Y.Q. Wu, Toroidal and annular Dehn fillings, Proc. London. Math. Soc. 78 (1999), 662-700. MR 1674841 (2000b:57029)

13.
C. McA. Gordon and Y.Q. Wu, Annular and boundary reducing Dehn fillings, Topology 39 (2000), 531-548. MR 1746907 (2001b:57033)

14.
C. McA. Gordon and Y.Q. Wu, Annular Dehn fillings, Comment. Math. Helv. 75 (2000), 430-456. MR 1746907 (2001b:57033)

15.
W. H. Jaco, and P. B. Shalen, Seifert fibered spaces in $ 3$-manifolds, Mem. Amer. Math. Soc. 21 (1979), no. 220. MR 0539411 (81c:57010)

16.
K. Johannson, Homotopy equivalences of $ 3$-manifolds with boundaries, Lecture Notes in Mathematics, 761, Springer, Berlin, (1979). MR 0551744 (82c:57005)

17.
H. Kneser, Geschlossene Flächen in dreidimensionale Mannigfaltigkeiten, Jahresber. Deutsch. Math.-Verein. 38 (1929) 248-260.

18.
S. Lee, Reducing and toroidal Dehn fillings on $ 3$-manifolds boundedy two tori, Math. Res. Lett. 13 (2006), 287-306.

19.
S. Lee, S. Oh and M. Teragaito, Dehn fillings and small surfaces, Proc. London Math. Soc. 92 (2006), no. 1, 203-223. MR 2192390

20.
W.B.R. Lickorish, A representation of orientable combinatorial $ 3$-manifolds, Ann. of Math. 76 (1962), 531-540. MR 0151948 (27:1929)

21.
J. Milnor, A unique decomposition theorem for $ 3$-manifolds, Amer. J. Math. 84 (1962), 1-7. MR 0142125 (25:5518)

22.
R. Qiu, Reducible Dehn surgery and annular Dehn surgery, Pacific J. Math. 192 (2000), 357-368. MR 1744575 (2001b:57036)

23.
W.P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. 6(1982), 357-381. MR 0648524 (83h:57019)

24.
A. Wallace, Modifications and cobounding manifolds, Canad. J. Math. 12 (1960) 503-528. MR 0125588 (23:A2887)

25.
Y.Q. Wu, Dehn fillings producing reducible manifolds and toroidal manifolds, Topology 37 (1998), 95-108. MR 1480879 (98j:57033)

26.
Y.Q. Wu, Sutured manifold hierarchies, essential laminations, and Dehn surgery, J. Diff. Geom. 48 (1998), 407-437. MR 1638025 (99h:57043)

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Additional Information:

Sangyop Lee
Affiliation: School of Mathematics, Korea Institute for Advanced Study, 207-43 Cheongryangri-dong, Dongdaemun-gu Seoul 130-722, Korea
Address at time of publication: Department of Mathematics, Seoul National University, Seoul 151-747, Korea
Email: slee@kias.re.kr

DOI: 10.1090/S0002-9947-06-03892-X
PII: S 0002-9947(06)03892-X
Keywords: Reducible, annular, Dehn filling
Received by editor(s): October 27, 2003
Received by editor(s) in revised form: October 20, 2004
Posted: August 15, 2006
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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