[Paper Review] On the tmf-resolution of Z

Agnès Beaudry, Mark Behrens+3 more|arXiv (Cornell University)|Sep 29, 2019

Homotopy and Cohomology in Algebraic Topology13 references4 citations

TL;DR

This paper computes the tmf-based Adams spectral sequence for the type 2 spectrum Z, establishing its E₁-page and calculating d₁-differentials modulo v₂-torsion. Using a novel low-dimensional calculation technique, it fully determines the spectral sequence in stems < 40, proving the collapse of the K(2)-local Adams-Novikov spectral sequence and enabling the computation of the homotopy groups of Z_{K(2)}, with implications for the telescope conjecture.

ABSTRACT

We study the tmf-based Adams spectral sequence for the type 2 spectrum Z. We establish the structure of the E_1-page of this spectral sequence, and compute the d_1-differential modulo v_2-torsion. We develop a technique for performing low dimensional calculations, and use this to compute the spectral sequence fully in stems < 40. We use this computation to prove that the K(2)-local Adams-Novikov spectral sequence for Z studied by the third author and Egger collapses, resulting in the computation of the homotopy groups of Z_{K(2)}. We discuss how these computations fit with the conjectural failure of the telescope conjecture.

Motivation & Objective

• To understand the tmf-based Adams spectral sequence for the type 2 spectrum Z.
• To determine the structure of the E₁-page of this spectral sequence.
• To compute d₁-differentials modulo v₂-torsion.
• To develop a technique for low-dimensional computations in tmf-based spectral sequences.
• To use these computations to establish the collapse of the K(2)-local Adams-Novikov spectral sequence for Z and compute π_*Z_{K(2)}.

Proposed method

• Construct the tmf-based Adams spectral sequence for the spectrum Z.
• Analyze the E₁-page using the structure of tmf and the homotopy groups of Z.
• Apply a new computational technique tailored for low-dimensional tmf-resolution calculations.
• Compute d₁-differentials modulo v₂-torsion using the spectral sequence's algebraic structure.
• Leverage the full computation in stems < 40 to deduce the collapse of the K(2)-local Adams-Novikov spectral sequence.
• Use the collapse to compute the homotopy groups of the K(2)-localization of Z.

Experimental results

Research questions

• RQ1 What is the structure of the E₁-page of the tmf-based Adams spectral sequence for the type 2 spectrum Z?
• RQ2 How do d₁-differentials behave in this spectral sequence modulo v₂-torsion?
• RQ3 Can a new technique for low-dimensional tmf-resolution computations be effectively applied to compute the spectral sequence in low stems?
• RQ4 Does the K(2)-local Adams-Novikov spectral sequence for Z collapse, and if so, what are the consequences for π_*Z_{K(2)}?
• RQ5 How do these computations relate to the conjectural failure of the telescope conjecture?

Key findings

• The E₁-page of the tmf-based Adams spectral sequence for Z is fully described in terms of its algebraic structure.
• d₁-differentials are computed modulo v₂-torsion, providing key information about the spectral sequence's behavior.
• A new technique for low-dimensional tmf-resolution calculations is successfully developed and applied.
• The spectral sequence is computed completely in stems less than 40, yielding explicit information about the filtration and differentials.
• The K(2)-local Adams-Novikov spectral sequence for Z collapses, enabling the complete computation of π_*Z_{K(2)} in the computed range.
• These results provide evidence consistent with the conjectural failure of the telescope conjecture.