The telescope conjecture at height 2 and the tmf resolution. Issue 4 (27th October 2021)

• Record Type: Journal Article
• Authors: Beaudry, Agnès; Behrens, Mark; Bhattacharya, Prasit; Culver, Dominic; Xu, Zhouli
• Is Part Of: Journal of topology. Volume 14:Issue 4(2021)
• Page Start: 1243
• Page End: 1320
• Publication Date: 2021-10-27
• Subjects: 55Q10 -- 55Q51 -- 55T15 (primary); Topology -- Periodicals; 514.05
• DOI: 10.1112/topo.12208
• Languages: English
• ISSNs: 1753-8416

Abstract

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.