Boardman's whole-plane obstruction group for Cartan-Eilenberg systems

TL;DR

This paper investigates the properties of Boardman's whole-plane obstruction group within the context of Cartan-Eilenberg systems, establishing key surjectivity and kernel isomorphism results related to spectral sequences.

Contribution

It demonstrates that the colim-lim interchange morphism is surjective and characterizes its kernel as the whole-plane obstruction group for Cartan-Eilenberg systems.

Findings

The colim-lim interchange morphism is surjective.

The kernel of this morphism is isomorphic to Boardman's whole-plane obstruction group.

Results apply to the spectral sequences derived from Cartan-Eilenberg systems.

Abstract

Each extended Cartan--Eilenberg system $(H, \partial)$ gives rise to two exact couples and one spectral sequence. We show that the canonical colim-lim interchange morphism associated to $H$ is a surjection, and that its kernel is isomorphic to Boardman's whole-plane obstruction group $W$, for each of the two exact couples.