Conley´s Spectral Sequence via the Sweeping Algorithm

In this article we consider a spectral sequence $(E^r; d^r)$ associated to a filtered Morse-Conley chain complex $(C; \Delta)$ where $\Delta$ is a connection matrix. The underlying motivation is to understand connection matrices under continuation. We show how the spectral sequence is completely determined by a family of connection matrices. This family is obtained by a sweeping algorithm for $\Delta$ over fields $F$ as well as over $Z$. This algorithm constructs a sequence of similar matrices $\Delta^0=\Delta$, $\Delta^1$, ..., where each matrix is related to the others via a change-of-basis matrix. Each matrix $\Delta^r$ over $F$ (resp., over $Z$) determines the vector space (resp., $Z$-module) $E^r$ and the differential $d^r$. We also prove the integrality of the final matrix $\Delta^R$ produced by the sweeping algorithm over $Z$ which is quite surprising, mainly because the intermediate matrices in the process may not have this property. Several other properties of the change-of-basis matrices as well as the intermediate matrices $\Delta^r$ are obtained. The sweeping algorithm and the computation of the spectral sequence $(E^r; d^r)$ are implemented in the software Mathematica.