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Cancellations for Circle-valued Morse Functions via Spectral Sequences

In this article, a spectral sequence analysis of a filtered Novikov complex $(\mathcal{N}_{\ast}(f),Δ)$ 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 $(\mathcal{N}_{\ast}(f),Δ)$ 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 $(\mathcal{N}_{\ast}(f),Δ)$ is encoded in a family of matrices $Δ^r$ produced by the Spectral Sequence Sweeping Algorithm (SSSA), which has as its initial input the differential $Δ$. 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 $f^r$.