Topological Methods in Nonlinear Analysis

Cancellations for circle-valued Morse functions via spectral sequences

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

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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 $f^r$.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 51, Number 1 (2018), 259-311.

Dates
First available in Project Euclid: 18 January 2018

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1516244425

Digital Object Identifier
doi:10.12775/TMNA.2017.047

Mathematical Reviews number (MathSciNet)
MR3784745

Zentralblatt MATH identifier
06887981

Citation

Lima, Dahisy V. de S.; Neto, Oziride Manzoli; de Rezende, Ketty A.; da Silveira, Mariana R. Cancellations for circle-valued Morse functions via spectral sequences. Topol. Methods Nonlinear Anal. 51 (2018), no. 1, 259--311. doi:10.12775/TMNA.2017.047. https://projecteuclid.org/euclid.tmna/1516244425


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