Conley's spectral sequence via the sweeping algorithm

Abstract

In this article we consider a spectral sequence ( E r , d r ) 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 Δ 0 = Δ , Δ 1 , … , where each matrix is related to the others via a change-of-basis matrix. Each matrix Δ 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 Δ 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 Δ r are obtained.