Hostname: page-component-f554764f5-fr72s Total loading time: 0 Render date: 2025-04-10T20:19:02.356Z Has data issue: false hasContentIssue false
  • English
  • Français

On pseudo-Anosov maps which extend over two handlebodies

Published online by Cambridge University Press:  20 January 2009

D. D. Long
D. D. Long
Affiliation:
Department of Mathematics, University of California, Santa Barbara, CA. 93106, USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that there is a pair of handlebodies H1 and H2 with common boundary F with the properties:

(a) There is no essential simple closed curve in F bounding a disc in both H1 and H2.

(b) Given any positive number, there are essential simple curves Cii = 1, 2 on F, bounding discs in Hi whose distance apart in the Hausdorff topology on F is less than this positive number.

Such an example has consequences for Heegaard splittings and recognising the 3-sphere.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 33 , Issue 2 , June 1990 , pp. 181 - 190
Copyright
Copyright © Edinburgh Mathematical Society 1990

References

REFERENCES

1.Bonahon, F., Cobordism of automorphisms of surfaces, Ann. Sci. École, Norm. Sup. (4), 16 (1983), 237–270.CrossRefGoogle Scholar
2.Casson, A. J., Automorphisms after Nielsen and Thurston (LMS Lecture Notes, 15, Cambridge, 1988).CrossRefGoogle Scholar
3.Casson, A. J. and Gordon, C. McA., A loop theorem for duality spaces and fibred ribbon knots, Invent. Math. 74 (1983), 119–139.CrossRefGoogle Scholar
4.Casson, A. J. and Long, D. D., Algorithmic compression of surface automorphisms, Invent. Math. 81 (1985), 295–303.CrossRefGoogle Scholar
5.Fathi, A. et al. , Travaux de Thurston sur les surfaces, Astérisque 66–67.Google Scholar
6.Goldsmith, D., Symmetric fibered links, Knots, groups and 3-manifolds (Ann. of Math. Studies 84, Princeton, 1975).Google Scholar
7.Hempel, J., 3-Manifolds (Ann. of Math. Studies 86, Princeton University Press, 1976).Google Scholar
8.Long, D. D., Cobordism of knot and surface automorphisms (Thesis, Cambridge University, 1983).Google Scholar
9.Long, D. D., Discs in compression bodies, Pacific J. Math. 122 (1986), 129–146.CrossRefGoogle Scholar
10.Long, D. D., Bounding laminations, Duke J. Math. 56 (1988), 1–16.CrossRefGoogle Scholar
11.Stallings, J., Constructions of fibred knots and links (Proc. Symp. in Pure Math. 32, Part II, 1978), 55–60.CrossRefGoogle Scholar