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Mathematics, Algebraic Topology
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.
