Filtration Shifting Maps and Differentials

Abstract

In this master's thesis we derive a connection between filtration shifts and differentials in a spectral sequence. We assume that the spectral sequence comes from a Cartan-Eilenberg system, and we develop a framework to fit the mapping cones of maps of filtered spectra or chain complexes into a sequence of Cartan-Eilenberg systems. Restricting to three-stage filtrations of the Cartan-Eilenberg systems, we give a complete description of this connection. Specifically, we show that a filtration shift leads to a non-zero differential in the spectral sequence associated to the mapping cone, and vice versa. We also give a slight generalisation of this result for longer filtrations, determining conditions at the level of the Cartan-Eilenberg systems that lets us reduce to the case of three-stage filtrations.