The telescope conjecture at height 2 and the tmf resolution

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 specific type 2 complex Z. We find that, analogous to the height 1 case, the E 1 -page of the tmf-resolution possesses a decomposition into a v 2 -periodic summand, and an Eilenberg–MacLane summand which consists of bounded v 2 -torsion. However, unlike the height 1 case, the E 2 -page of the tmf-resolution exhibits unbounded v 2 -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 v 2 -torsion: either the unbounded v 2 -torsion kills itself off in the spectral sequence, and the telescope conjecture is true, or it persists to form v 2 -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.