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Algorithmic recognition of 3-manifolds
Author(s):
Abigail
Thompson
Journal:
Bull. Amer. Math. Soc.
35
(1998),
57-66.
MSC (1991):
Primary 57M40
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Additional information
Abstract:
This article discusses recent progress in algorithmically classifying 3-manifolds by homeomorphism type.
References:
- [AMR]
- I. R. Aitchison, S. Matsumoto and J. H. Rubinstein, Immersed surfaces in the figure-8 knot complement, preprint.
- [CGLS]
- M. Culler, C. McA. Gordon, J. Luecke, and P. Shalen, Dehn surgery on knots, Ann. of Math. 125 (1987), 237-300. MR 88a:57026
- [F]
- M. Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982), 357-453. MR 84b:57006
- [G]
- D. Gabai, Foliations and the topology of 3-manifolds III, J. Differential Geom. 26 (1987), 479-536. MR 89a:57014b
- [GL]
- C. McA. Gordon and J. Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989), 371-415. MR 90a:57006a
- [Gr]
- M. Grayson, Shortening embedded curves, Ann. of Math. 129 (1989), 71-111. MR 90a:53050
- [H1]
- W. Haken, Theorie der Normalflachen, Acta. Math. 105 (1961), 245-375. MR 25:4519a
- [H2]
- -, Some results on surfaces in 3-manifolds, Studies in Modern Topology, Math. Assoc. Amer., Prentice-Hall, Englewood Cliffs, NJ, 1968, pp. 39-98. MR 36:7118
- [HLP]
- J. Hass, J. Lagarias, and N. Pippinger, The computational complexity of knot and link algorithms, preprint.
- [He]
- G. Hemion, The classification of knots and 3-dimensional spaces, Oxford Univ. Press, 1992. MR 94g:57015
- [Hp]
- J. Hempel, 3-manifolds, Ann. of Math. Stud., no. 86, Princeton Univ. Press, Princeton, NJ, 1976. MR 54:3702
- [HU]
- J. Hopcroft and J. Ullman, Introduction to automata theory, languages and computation, Addison Wesley, Reading, MA, 1979. MR 83j:68002
- [JO]
- W. Jaco and U. Oertel, An algorithm to decide if a 3-manifold is a Haken manifold, Topology 23 (1984), 195-209. MR 85j:57014
- [JR]
- W. Jaco and J. H. Rubinstein, A piecewise linear theory of minimal surfaces in 3-manifolds, J. Differential Geom. 27 (1988). MR 88a:57023
- [JT]
- W. Jaco and J. L. Tollefson, Algorithms for the complete decomposition of a closed 3-manifold, Illinois J. Math. 39 (1995), 358-406. MR 97a:57014
- [J]
- K. Johannsen, Classification problems in low-dimensional topology, geometric and algebraic topology, Banach Center Publ., vol. 18, PWN, Warsaw, 1986.
- [LS]
- L. Lusternik and L. Schnirelman, Sur le probleme de trois geodesiques fermees sur les surface de genre
 , C. R. Acad. Sci. Paris 189 (1929). - [M]
- S. V. Matveev, A recognition algorithm for the 3-dimensional sphere (after A. Thompson), Mat. Sb. 186 (1995), 69-84; English transl. Math. USSR-Sb 186 (1995). MR 96g:57016
- [Mo]
- E. E. Moise, Affine structures in 3-manifolds, Ann. of Math. 55 (1952). MR 13:765a; MR 13:765b; MR 13:765c
- [Ra]
- R. Rannard, Computational enumeration of immersed normal surfaces in the figure eight knot complement, preprint.
- [R1]
- J. H. Rubinstein, An algorithm to recognize the 3-sphere, Proc. ICM (Zurich, 1994), Vols. 1, 2, Birkhäuser, Basel, 1995, pp. 601-611. MR 97e:57011
- [R2]
- -, The solution to the recognition problem for
 , Lectures, Haifa, Israel, May 1992. - [R3]
- -, Polyhedral minimal surfaces, Heegaard splittings and decision problems for 3-manifolds, Proc. Georgia Topology Conf., Amer. Math. Soc./Intl. Press, 1993.
- [S]
- H. Schubert, Bestimmung der Primfaktorzerlegung von Verkettungen, Math. Z. 76 (1961), 116-148. MR 25:4519b
- [Sm]
- S. Smale, Generalized Poincaré's conjecture in dimensions greater than four, Ann. of Math. (2) 74 (1961), 391-406. MR 25:580
- [SS]
- S. Simon and F. Smith, On the existence of embedded minimal 2-spheres in the 3-sphere, endowed with an arbitrary metric, preprint.
- [St]
- M. Stocking, Almost normal surfaces in 3-manifolds, Ph.D. thesis, UC Davis, 1996.
- [T1]
- A. Thompson, Thin position and the recognition problem for
 , Math. Res. Lett. 1 (1994), 613-630. MR 95k:57015 - [T2]
- -, Thin position and bridge number for knots in the 3-sphere, Topology 36 (1997), 505-507. MR 97m:57013
- [W]
- F. Waldhausen, Recent results on sufficiently large 3-manifolds, Proc. Sympos. Pure Math., vol. 32, Amer. Math., Soc., Providence, RI, 1978. MR 80e:57010
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Additional Information:
Abigail
Thompson
Affiliation:
Department of Mathematics, University of California Davis, Davis, California 95616-5224
Email:
thompson@math.ucdavis.edu
DOI:
10.1090/S0273-0979-98-00738-1
PII:
S 0273-0979(98)00738-1
Received by editor(s):
October 1, 1997
Copyright of article:
Copyright
1998,
American Mathematical Society
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