Skip to main content
×
×
Home

Incompressible surfaces and the topology of 3-dimensional manifolds

  • Iain R. Aitchison (a1) and J. Hyam Rubinstein (a1)
Abstract

Existence and properties of incompressible surfaces in 3-dimensional manifolds are surveyed. Some conjectures of Waldhausen and Thurston concerning such surfaces are stated. An outline is given of the proof that such surfaces can be pulled back by non-zero degree maps between 3-manifolds. The effect of surgery on immersed, incompressible surfaces and on hierarchies is discussed. A characterisation is given of the immersed, incompressible surfaces previously studied by Hass and Scott, which arise naturally with cubings of non-positive curvature.

Copyright
References
Hide All
[1]Aitchison, I. R., Lumsden, E. and Rubinstein, J. H., ‘Cusp structure of alternating links’, Invent. Math. 109 (1992), 473494.
[2]Aitchison, I. R. and Rubinstein, J. H., ‘Geodesic surfaces in knot complements’, preprint, University of Melbourne, 1990.
[3]Aitchison, I. R. and Rubinstein, J. H., ‘An introduction to polyhedral metrics of non-positive curvature on 3-manifolds’, in: Geometry of Low-Dimensional Manifolds: 2 (eds. Donaldson, S. K. and Thomas, C. B.), London Math. Soc. Lecture Notes 151 (Cambridge University Press, Cambridge, 1990) pp. 127161.
[4]Aitchison, I. R. and Rubinstein, J. H., ‘Canonical surgery on alternating link diagrams’, in: Knots 90 (ed. Kawauchi, A.) (de Gruyter, Berlin, 1992) pp. 543558.
[5]Aitchison, I. R. and Rubinstein, J. H., ‘Combinatorial cubings, cusps and the dodecahedral knots’, in: Topology 90 (eds. Apanasov, , Neumann, , Reid, and Siebenmann, ) (de Gruyter, Berlin, 1992) pp. 1726.
[6]Culler, M. and Shalen, P., ‘Varieties of group representations and splittings of 3-manifolds’, Ann. of Math. 117 (1983), 109146.
[7]Epstein, D., ‘Periodic flows on three-manifolds’, Ann. of Math. 95 (1972), 6682.
[8]Freedman, M., Hass, J. and Scott, P., ‘Least area incompressible surfaces in 3-manifolds’, Invent. Math. 71 (1983), 609642.
[9]Gabai, D., ‘Homotopy hyperbolic 3-manifolds are virtually hyperbolic’, preprint, 1992.
[10]Gordon, C. and Luecke, J., ‘Knots are determined by their complements’, J. Amer. Math. Soc. 2 (1989), 371415.
[11]Gromov, M., ‘Hyperbolic groups’, in: Essays in Group Theory (ed. Gersten, S.), MSRI publications 8 (Springer, Berlin, 1987) pp. 75264.
[12]Haken, W., ‘Theorie der Normal Flächen’, Acta Math. 105 (1961), 245375.
[13]Haken, W., ‘Über das Homöomorphieproblem der 3-Mannigfaltigkeiten I’, Math. Z. 80 (1962), 89120.
[14]Haken, W., ‘Some results on surfaces in 3-manifolds’, in: Studies in Modern Topology (Math. Assoc. Amer., Washington D.C., 1968) pp. 3498.
[15]Hass, J. and Scott, P., ‘Homotopy equivalence and homeomorphism of 3-manifolds’, Topology 31 (1992), 493517.
[16]Hatcher, A., ‘On the boundary curves of incompressible surfaces’, Pacific J. Math. 99 (1982), 373377.
[17]Hemion, G., ‘On the classification of homeomorphisms of 2-manifolds and the classification of 3-manifolds’, Acta Math. 142 (1979), 123155.
[18]Hempel, J., 3-Manifolds, Annals of Math. Studies 86 (Princeton University Press, Princeton, 1976).
[19]Hempel, J., ‘Homology of coverings’, Pacific J. Math. 112 (1984), 83113.
[20]Hempel, J., ‘Residual finiteness for 3-manifolds’, in: Combinatorial group theory and topology, volume 111 of Annals of Math. Studies (Princeton University Press, Princeton, 1987) pp. 379396.
[21]Jaco, W., Lectures on three-manifold topology, Conf. Board of Math. Sci. 43 (American Math. Society, Providence, 1980).
[22]Jaco, W. and Rubinstein, J. H., ‘PL minimal surfaces in 3-manifolds’, J. Differential Geom. 27 (1988), 493524.
[23]Jaco, W. and Shalen, P., Seifert Fibered Spaces in 3-Manifolds, Mem. Amer. Math. Soc. 220 (1980).
[24]Johannson, K., Homotopy Equivalences of 3-manifolds with Boundary, Lecture Notes in Mathematics 761 (Springer-Verlag, Berlin, 1979).
[25]Kneser, H., ‘Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten’, Jahresber. Deutsch Math.-Verein 38 (1929), 248260.
[26]Lickorish, W. B. R., ‘A representation of orientable combinatorial 3-manifolds’, Ann. of Math. 76 (1962), 531540.
[27]Long, D., ‘Immersions and embeddings of totally geodesic surfaces’, Bull. London Math. Soc. 19 (1987), 481484.
[28]Markov, A. A., ‘Unsolvability of the problem of homeomorphy’, in: Proceedings of the International Congress of Mathematicians 1958 (ed. Todd, J. A.) (Cambridge University Press, Cambridge, 1960) pp. 300306.
[29]Millson, J., ‘On the first Betti number of a constant negatively curved manifold’, Ann. of Math. 104 (1976), 235247.
[30]Milnor, J., ‘A unique factorisation theorem for 3-manifolds’, Amer. J. Math. 84 (1962), 17.
[31]Moise, E., ‘Affine structures in 3-manifolds V. the triangulation theorem and hauptvermutung’, Ann. of Math. 56 (1952), 96114.
[32]Mostow, G., ‘Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms’, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 53104.
[33]Orlik, P., Seifert Manifolds, Lecture Notes in Mathematics 291 (Springer-Verlag, Berlin, 1972).
[34]Papakyriakopoulos, C., ‘On Dehn's Lemma and the asphericity of knots’, Ann. of Math. 66 (1957), 126.
[35]Papakyriakopoulos, C., ‘On solid tori’, Proc. London Math. Soc. 7 (1957), 281299.
[36]Rubinstein, J. H. and Swarup, G. A., ‘On Scott's core theorem’, Bull. London Math. Soc. 22 (1990), 495498.
[37]Schubert, H., ‘Bestimmung der Primfaktorzerlegung von Verkettungen’, Math. Zeit. 76 (1961), 116148.
[38]Scott, P., ‘Compact submanifolds of 3-manifolds’, J. London Math. Soc. (2) 7 (1973), 246250.
[39]Scott, P., ‘The geometries of 3-manifolds’, Bull. London Math. Soc. 15 (1983), 401487.
[40]Seifert, H., ‘Topologie dreidimensionalen gefaserter Räume’, Acta Math. 60 (1933), 147238.
[41]Skinner, A., The word problem in the fundamental groups of a class of three dimensional manifolds (Ph.D. Thesis, University of Melbourne, 1991).
[42]Stallings, J., ‘On the loop theorem’, Ann. of Math. 72 (1960), 1219.
[43]Stallings, J., ‘On fibering certain 3-manifolds’, in: Topology of 3-Manifolds (ed. Fort, M.) (Prentice-Hall, 1962) pp. 95100.
[44]Swarup, G. A., ‘On a theorem of Johannson’, J. London Math. Soc. 18 (1978), 560562.
[45]Thurston, W., The geometry and topology of 3-manifolds (Lecture notes, Princeton University, 1978).
[46]Thurston, W., ‘Three dimensional manifolds, Kleinian groups and hyperbolic geometry’, Bull. Amer. Math. Soc. 6 (1982), 357381.
[47]Thurston, W., ‘Hyperbolic structures on 3-manifolds I: Deformations of acylindrical manifolds’, Ann. of Math. 124 (1986), 203246.
[48]Waldhausen, F., ‘On irreducible 3-manifolds which are sufficiently large’, Ann. of Math. 87 (1968), 5688.
[49]Waldhausen, F., ‘The word problem in fundamental groups of sufficiently large 3-manifolds’, Ann. of Math. 88 (1968), 272280.
[50]Waldhausen, F., ‘On the determination of some bounded 3-manifolds by their fundamental groups alone’, in: Proc. of Inter. Sym. on Topology (Yugoslavia, Beograd, 1969) pp. 331332.
[51]Wu, Y-Q., ‘Incompressibility of surfaces in surgered 3-manifolds’, Topology 31 (1992), 271279.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed