Conley's spectral sequence via the sweeping algorithm

In this article we consider a spectral sequence (Er,dr) associated to a filtered Morse–Conley chain complex (C,Δ), where Δ 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 Δ over fields F as well as over Z. This algorithm constructs a sequence of similar matrices Δ0=Δ,Δ1,… , where each matrix is related to the others via a change-of-basis matrix. Each matrix Δr over F (resp., over Z) determines the vector space (resp., Z-module) Er and the differential dr. We also prove the integrality of the final matrix ΔR 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 Δr are obtained.