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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
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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

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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