Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-12-05T02:35:32.210Z Has data issue: false hasContentIssue false
  • English
  • Français

Gradient-like flows on 3-manifolds

Published online by Cambridge University Press:  19 September 2008

K. A. de Rezende
K. A. de Rezende
Affiliation:
Departamento de Matemática, Universidade Estadual de Campinas, 13081–970 Campinas, Sāo Paulo, Brazil
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we determine properties that a Lyapunov graph must satisfy for it to be associated with a gradient-like flow on a closed orientable three-manifold. We also address the question of the realization of abstract Lyapunov graphs as gradient-like flows on three-manifolds and as a byproduct we prove a partial converse to the theorem which states the Morse inequalities for closed orientable three-manifolds. We also present cancellation theorems of non-degenerate critical points for flows which arise as realizations of canonical abstract Lyapunov graphs.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 13 , Issue 3 , September 1993 , pp. 557 - 580
Copyright
Copyright © Cambridge University Press 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[B]Brakes, W. R.. Sewing-up link exteriors. Low Dimensional Topology. London Mathematical Society Lecture Note Series 48. Bangor, 1979. pp 27–37.Google Scholar
[C]Conley, C.. Isolated Invariant Sets and the Morse Index. CBMS Regional Conf. Series in Math. 38. American Mathematical Society: Providence, RI, 1978.CrossRefGoogle Scholar
[dR]de Rezende, K.. Smale flows on the three-sphere. Trans. Amer. Math. Soc. 303 (1987), 283–310.CrossRefGoogle Scholar
[dR, F]de Rezende, K. & Franzosa, R.. Lyapunov graphs and flows on surfaces. Trans. Amer. Math. Soc. To appear.Google Scholar
[F1]Franks, J.. Non-singular Smale flows on S 3. Topology 24 (1985), 265–282.CrossRefGoogle Scholar
[F2]Franks, J.. Homology and Dynamical Systems. CBMS Regional Conf. Series in Math. 49. American Mathematical Society: Providence, RI, 1982.Google Scholar
[FKV]Fomenko, A. T., Kuznetsov, V. & Volodin, I.. The problem of discriminating algorithmically the standard three-dimensional sphere. Russian Math. Surveys 29(5) (1974), 71–172.Google Scholar
[H]Hempel, J.. 3-Manifolds, Annals of Mathematical Studies 86. Princeton University Press: Princeton, 1976.Google Scholar
[M1]Milnor, J.. Lectures on the h-cobordism Theorem. Princeton Mathematical Notes. Princeton University Press: Princeton, 1965.CrossRefGoogle Scholar
[M2]Milnor, J.. A unique decomposition theorem for 3-manifolds. Amer. J. Math. 84 (1962), 1–7.CrossRefGoogle Scholar
[S1]Smale, S.. On gradient dynamical systems. Ann. Math. 74 (1961), 199–206.CrossRefGoogle Scholar
[S2]Smale, S.. Generalized Poincaré's Conjecture in dimensions greater than four. Ann. Math. 74 (1961), 391–406.CrossRefGoogle Scholar
[Sm]Smoller, J.. Shock Waves and Reaction Diffusion Equations. Springer: Berlin, 1983.CrossRefGoogle Scholar