Cancellations for circle-valued Morse functions via spectral sequences

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



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 $f^r$.


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

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

P. Bartlomiejczyk, The Conley index and spectral sequences, Topol. Methods Nonlinear Anal. 25 (2005), 195–203.

P. Bartlomiejczyk, Spectral sequences and detailed connection matrices, Topol. Methods Nonlinear Anal. 34 (2009), 187–200.

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.

J.F. Davis and P. Kirk, Lecture Notes in Algebraic Topology, Graduate Studies in Mathematics, vol. 35, American Mathematical Society, Providence, 2001.

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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