[Paper Review] Total Cofibres of Diagrams of Spectra

Thomas Huettemann|ArXiv.org|Aug 22, 2005

Homotopy and Cohomology in Algebraic Topology2 references3 citations

TL;DR

This paper introduces the total cofibre construction for diagrams of spectra indexed by posets, defining it as the homotopy cofibre of the inclusion of homotopy colimits over a subposet. It establishes a stable equivalence between the homotopy limit and a mapping spectrum into the total cofibre when the poset pair satisfies a condition on stable cohomotopy, generalizing spectral sequences for sheaf cohomology and extending results to spectra and topological spaces via Bousfield-Kan methods.

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 Γ(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 Γ(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 Γ(Y) with E_2-term involving higher derived inverse limits of π_* (Y), generalising earlier constructions for space-valued diagrams indexed by the face lattice of a polytope.

Motivation & Objective

• To define and study the total cofibre of a diagram of spectra indexed by a poset pair $\mathcal{D} \subset \mathcal{C}$, generalizing constructions from space-valued diagrams.
• To construct a comparison map from the homotopy limit of a diagram to a mapping spectrum into the total cofibre.
• To characterize precisely which poset pairs $\mathcal{D} \subset \mathcal{C}$ make this comparison map a stable equivalence.
• To generalize the Bousfield-Kan spectral sequence for homotopy limits to the setting of total cofibres, especially for diagrams on polytopal complexes.
• To establish a stable equivalence between total cofibres and homotopy limits when $|\mathcal{C}|$ is a PL ball and $|\mathcal{D}|$ its boundary, up to $m$-fold looping.

Proposed method

• Define the total cofibre $\Gamma(Y)$ as the homotopy cofibre of the map $\mathrm{hocolim}_{\mathcal{D}}(Y) \to \mathrm{hocolim}_{\mathcal{C}}(Y)$ for a diagram $Y$ of spectra over a poset $\mathcal{C}$ with subposet $\mathcal{D}$.
• Construct a comparison map $\hat{\Gamma}_Y: \mathrm{holim}_{\mathcal{C}}(Y) \to \hom(Z, \hat{\Gamma}(Y))$, where $Z$ is a simplicial set derived from $\mathcal{C}$ and $\mathcal{D}$, and $\hat{\Gamma}(Y)$ is a fibrant replacement of $\Gamma(Y)$.
• Characterize the stable equivalence of $\hat{\Gamma}_Y$ in terms of the stable cohomotopy of the space $Z$, showing that $\hat{\Gamma}_Y$ is a stable equivalence precisely when $\pi_s^n(Z) = 0$ for $n < 0$.
• Use the Bousfield-Kan spectral sequence for homotopy limits to construct a spectral sequence for $\pi_*(\Gamma(Y))$ with $E^2$-term involving higher derived inverse limits $\lim^p(\pi_*(Y))$.
• Prove that for $\mathcal{C}$ a finite polytopal complex with $|\mathcal{C}| \cong B^m$ and $|\mathcal{D}| \cong S^{m-1}$, the comparison map becomes a stable equivalence after $m$-fold looping.
• Apply the result to diagrams of topological spectra and to space-valued diagrams via the Eilenberg-MacLane spectrum $H\mathbb{Z}$, recovering and generalizing earlier spectral sequences.

Experimental results

Research questions

• RQ1 Under what conditions on the poset pair $\mathcal{D} \subset \mathcal{C}$ is the comparison map from the homotopy limit to the mapping spectrum into the total cofibre a stable equivalence?
• RQ2 How does the total cofibre construction for spectra relate to classical homotopy limits in stable homotopy theory?
• RQ3 Can the Bousfield-Kan spectral sequence for homotopy limits be adapted to compute the homotopy groups of total cofibres?
• RQ4 What is the relationship between total cofibres and sheaf cohomology in algebraic geometry, particularly in the context of toric varieties?
• RQ5 What spectral sequence governs the homotopy groups of the total cofibre for diagrams on polytopal complexes with ball-like posets?

Key findings

• The comparison map $\hat{\Gamma}_Y: \mathrm{holim}_{\mathcal{C}}(Y) \to \hom(Z, \hat{\Gamma}(Y))$ is a stable equivalence if and only if the stable cohomotopy groups $\pi_s^n(Z)$ vanish for all $n < 0$, where $Z$ is the simplicial set associated to the poset pair $\mathcal{D} \subset \mathcal{C}$.
• For $\mathcal{C}$ a finite polytopal complex with $|\mathcal{C}| \cong B^m$ and $|\mathcal{D}| \cong S^{m-1}$, the total cofibre $\Gamma(Y)$ and the homotopy limit $\mathrm{holim}_{\mathcal{C}}(Y)$ are stably equivalent up to $m$-fold looping.
• A strongly convergent spectral sequence $E^2_{p,q} = \lim^p(\pi_q Y) \Rightarrow \pi_{m+q-p}(\Gamma(Y))$ computes the homotopy groups of the total cofibre for diagrams on $m$-dimensional polytopal balls.
• When $Y = X \wedge H\mathbb{Z}$ for a diagram $X: \mathcal{C} \to \mathrm{Top}_*$, the spectral sequence becomes $E^2_{p,q} = \lim^p(\tilde{H}_q(X;\mathbb{Z})) \Rightarrow \tilde{H}_{m+q-p}(\Gamma(X);\mathbb{Z})$, generalizing earlier results for polytopes.
• The total cofibre construction for spectra is a stable analogue of sheaf cohomology, replacing global sections and derived functors in algebraic geometry.
• The spectral sequence for $\Gamma(Y)$ via $\lim^p(\pi_q Y)$ provides a homotopical interpretation of higher derived inverse limits in terms of stable homotopy theory.