Total Cofibres of Diagrams of Spectra

TL;DR

This paper introduces the total cofibre of a diagram of spectra indexed by a poset, providing a comparison map to homotopy limits, characterising when they are equivalent, and applying this to spectral sequences for homotopy groups.

Contribution

It defines the total cofibre for diagrams of spectra, constructs a comparison map to homotopy limits, and characterises when they are stably equivalent, extending previous spectral sequence results.

Findings

The comparison map is a stable equivalence for certain poset pairs.

The total cofibre agrees with homotopy limits up to looping in specific cases.

A spectral sequence for homotopy groups of the total cofibre is constructed.

Abstract

If Y is a diagram of spectra indexed by an arbitrary poset C together with a specified sub-poset D, we define the total cofibre \Gamma (Y) of Y as the strict cofibre of the map from hocolim_D (Y) to hocolim_C (Y). We construct a comparison map from the homotopy limit of Y to a looping of a fibrant replacement of Gamma (Y), and characterise those poset pairs (C,D) for which this comparison map is a stable equivalence. The characterisation is given in terms of stable cohomotopy of spaces related to C and D. For example, if C is a finite polytopal complex with underlying space an m-ball with boundary sphere D, then holim_C (Y) and \Gamma(Y) agree up to m-fold looping and up to stable equivalence. As an application of the general result we give a spectral sequence for the homotopy groups of \Gamma(Y) with E_2-term involving higher derived inverse limits of \pi_* (Y), generalising earlier constructions for space-valued diagrams indexed by the face lattice of a polytope.