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    da Cruz, Ricardo Nogueira Mello, Margarida Pinheiro and de Rezende, Ketty Abaroa 2005. Realizability of the Morse polytope. Qualitative Theory of Dynamical Systems, Vol. 6, Issue. 1, p. 59.

  • Ergodic Theory and Dynamical Systems, Volume 13, Issue 3
  • September 1993, pp. 557-580

Gradient-like flows on 3-manifolds

  • K. A. de Rezende (a1)
  • DOI:
  • Published online: 01 September 2008

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.

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[C]C. Conley . Isolated Invariant Sets and the Morse Index. CBMS Regional Conf. Series in Math. 38. American Mathematical Society: Providence, RI, 1978.

[dR]K. de Rezende . Smale flows on the three-sphere. Trans. Amer. Math. Soc. 303 (1987), 283310.

[F1]J. Franks . Non-singular Smale flows on S3. Topology 24 (1985), 265282.

[F2]J. Franks . Homology and Dynamical Systems. CBMS Regional Conf. Series in Math. 49. American Mathematical Society: Providence, RI, 1982.

[M1]J. Milnor . Lectures on the h-cobordism Theorem. Princeton Mathematical Notes. Princeton University Press: Princeton, 1965.

[M2]J. Milnor . A unique decomposition theorem for 3-manifolds. Amer. J. Math. 84 (1962), 17.

[S1]S. Smale . On gradient dynamical systems. Ann. Math. 74 (1961), 199206.

[S2]S. Smale . Generalized Poincaré's Conjecture in dimensions greater than four. Ann. Math. 74 (1961), 391406.

[Sm]J. Smoller . Shock Waves and Reaction Diffusion Equations. Springer: Berlin, 1983.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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