The telescope conjecture at height 2 and the tmf resolution

TL;DR

This paper investigates the height 2 telescope conjecture at prime 2 using tmf-resolutions, revealing complex torsion behavior and providing new computational tools for homotopy groups, thus advancing understanding of chromatic phenomena.

Contribution

It introduces a detailed analysis of the tmf-resolution at height 2, highlighting differences from height 1 and connecting torsion behavior to the conjecture's validity.

Findings

E1-page decomposes into v2-periodic and Eilenberg-MacLane summands.

E2-page exhibits unbounded v2-torsion, unlike height 1 case.

Proves the collapse of the E(2)-local Adams-Novikov spectral sequence for Z.

Abstract

Mahowald proved the height 1 telescope conjecture at the prime 2 as an application of his seminal work on bo-resolutions. In this paper we study the height 2 telescope conjecture at the prime 2 through the lens of tmf-resolutions. To this end we compute the structure of the tmf-resolution for a specifc type 2 complex Z. We find that, analogous to the height 1 case, the E1-page of the tmf-resolution possesses a decomposition into a v2-periodic summand, and an Eilenberg-MacLane summand which consists of bounded v2-torsion. However, unlike the height 1 case, the E2-page of the tmf-resolution exhibits unbounded v2-torsion. We compare this to the work of Mahowald-Ravenel-Shick, and discuss how the validity of the telescope conjecture is connected to the fate of this unbounded v2-torsion: either the unbounded v2-torsion kills itself off in the spectral sequence, and the telescope conjecture is true, or it persists to form v2-parabolas and the telescope conjecture is false. We also study how to use the tmf-resolution to effectively give low dimensional computations of the homotopy groups of Z. These computations allow us to prove a conjecture of the second author and Egger: the E(2)-local Adams-Novikov spectral sequence for Z collapses.