Hostname: page-component-7c8c6479df-7qhmt Total loading time: 0 Render date: 2024-03-28T07:45:03.131Z Has data issue: false hasContentIssue false

Gradient-like flows on 3-manifolds

Published online by Cambridge University Press:  19 September 2008

K. A. de Rezende
Affiliation:
Departamento de Matemática, Universidade Estadual de Campinas, 13081–970 Campinas, Sāo Paulo, Brazil

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
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 2737.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.Google Scholar
[dR]de Rezende, K.. Smale flows on the three-sphere. Trans. Amer. Math. Soc. 303 (1987), 283310.Google 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), 265282.Google 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), 71172.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.Google Scholar
[M2]Milnor, J.. A unique decomposition theorem for 3-manifolds. Amer. J. Math. 84 (1962), 17.Google Scholar
[S1]Smale, S.. On gradient dynamical systems. Ann. Math. 74 (1961), 199206.Google Scholar
[S2]Smale, S.. Generalized Poincaré's Conjecture in dimensions greater than four. Ann. Math. 74 (1961), 391406.Google Scholar
[Sm]Smoller, J.. Shock Waves and Reaction Diffusion Equations. Springer: Berlin, 1983.Google Scholar