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Topological Methods in Nonlinear Analysis

Cancellations for circle-valued Morse functions via spectral sequences
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Cancellations for circle-valued Morse functions via spectral sequences

Authors

  • Dahisy V. de S. Lima
  • Oziride Manzoli Neto
  • Ketty A. de Rezende
  • Mariana R. da Silveira

Keywords

Cancellation, circle-valued functions, Novikov complex, spectral sequence

Abstract

A spectral sequence analysis of a filtered Novikov complex $(\mathcal{N}_{\ast}(f),\Delta)$ over $\mathbb{Z}((t))$ is developed with the goal of obtaining results relating the algebraic and dynamical settings. Specifically, the unfolding of a spectral sequence of $(\mathcal{N}_{\ast}(f),\Delta)$ and the cancellation of its modules is associated to a one parameter family of circle-valued Morse functions on a surface and the dynamical cancellations of its critical points. The data of a spectral sequence computed for $(\mathcal{N}_{\ast}(f),\Delta)$ is encoded in a family of matrices $\Delta^r$ produced by the Spectral Sequence Sweeping Algorithm (SSSA), which has as its initial input the differential $\Delta$. As one ``turns the pages'' of the spectral sequence, differentials which are isomorphisms produce cancellation of pairs of modules. Corresponding to these cancellations, a family of circle-valued Morse functions $f^r$ is obtained by successively removing the corresponding pairs of critical points of $f$. We also keep track of all dynamical information on the birth and death of connecting orbits between consecutive critical points, as well as periodic orbits that arise within a family of negative gradient flows associated to $f^r$.

References

A. Banyaga and D. Hurtubise, Lecture on Morse Homology, Kluwer Texts in the Mathematical Sciences, vol. 29, Kluwer Academic Publishers Group, Dordrecht, 2004.

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M.A. Bertolim, D.V.S. Lima, M.P. Mello, K.A. de Rezende and M.R. da Silveira, A global two-dimensional version of Smale’s cancellation theorem via spectral sequences, Ergodic Theory and Dynamical Systems 36 (2016), 1795–1838.

M.A. Bertolim, D.V.S. Lima, M.P. Mello, K.A. de Rezende and M.R. da Silveira, An algorithmic approach to algebraic and dynamical cancellations associated to a spectral sequence, arXiv:1408.6286 [math.DS].

O. Cornea, K.A. de Rezende and M.R. da Silveira, Spectral sequences in Conley’s theory, Ergodic Theory Dynam. Systems 30 (2010), 1009–1054.

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R.D. Franzosa, K.A. de Rezende and M.R. da Silveira, Continuation and bifurcation associated to the dynamical spectral sequence, Ergodic Theory Dynam. Systems 34 (2014), 1849–1887.

F. Latour, Existence de 1-formes fermées non singulières daus une classe de cohomologie de De Rham, Inst. Hautes Études Scient. Publ. Math. 80 (1994), 135–194.

M.P. Mello, K.A. de Rezende and M.R. da Silveira, Conley’s spectral sequences via the sweeping algorithm, Topology Appl. 157 (2010), 2111–2130.

J. Milnor, Lectures on the h-Cobordism, Princeton University Press, New Jersey, 1965.

A.V. Pajitnov, Circle-Valued Morse Theory, De Gruyter Studies in Mathematics, vol. 32, Walter de Gruyter, Berlin, 2006.

D.A. Salamon, Morse theory, the Conley index and the Floer homology, Bull. London Math. Soc. 22 (1990), 113–240.

E. Spanier, Algebraic Topology, McGraw–Hill, New York, 1966.

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Published

2017-12-09

How to Cite

1.
LIMA, Dahisy V. de S., NETO, Oziride Manzoli, DE REZENDE, Ketty A. and DA SILVEIRA, Mariana R. Cancellations for circle-valued Morse functions via spectral sequences. Topological Methods in Nonlinear Analysis. Online. 9 December 2017. Vol. 51, no. 1, pp. 259 - 311. [Accessed 2 July 2025].
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Vol 51, No 1 (March 2018)

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