The goal of this paper is to develop some of the machinery necessary for doing K.2/-local computations in the stable homotopy category using duality resolutions at the prime p D 2. The Morava stabilizer group S 2 admits a surjective homomorphism to Z 2 whose kernel we denote by S 1 2 . The algebraic duality resolution is a finite resolution of the trivial Z 2 OEOES 1 2 -module Z 2 by modules induced from representations of finite subgroups of S 1 2 . Its construction is due to Goerss, Henn, Mahowald and Rezk. It is an analogue of their finite resolution of the trivial Z 3 OEOEG 1 2 -module Z 3 at the prime p D 3. The construction was never published and it is the main result in this paper. In the process, we give a detailed description of the structure of Morava stabilizer group S 2 at the prime 2. We also describe the maps in the algebraic duality resolution with the precision necessary for explicit computations. 55Q45; 55T99, 55P60

Let V (0) be the mod 2 Moore spectrum and let C be the supersingular elliptic curve over F 4 defined by the Weierstrass equation y 2 +y = x 3 . Let F C be its formal group law and E C be the spectrum classifying the deformations of F C . The group of automorphisms of F C , which we denote by S C , acts on E C . Further, S C admits a surjective homomorphism to Z 2 whose kernel we denote by S 1 C . The cohomology of S 1 C with coefficients in (E C ) * V (0) is the E 2 -term of a spectral sequence converging to the homotopy groups of E hS 1. In this paper, we use the algebraic duality resolution spectral sequence to compute an associated graded for H * (S 1 C ; (E C ) * V (0)). These computations rely heavily on the geometry of elliptic curves made available to us at chromatic level 2.

This paper contains an overview of background from stable homotopy theory used by Freed-Hopkins in their work on invertible extended topological field theories. We provide a working guide to the stable homotopy category, to the Steenrod algebra and to computations using the Adams spectral sequence. Many examples are worked out in detail to illustrate the techniques.

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 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.

We show that the strongest form of Hopkins' chromatic splitting conjecture, as stated by Hovey in [12], cannot hold at chromatic level n = 2 at the prime p = 2. More precisely, for V(0) the mod 2 Moore spectrum, we prove that π k L 1 L K(2) V(0) is not zero when k is congruent to −3 modulo 8. We explain how this contradicts the decomposition of L 1 L K(2) S predicted by the chromatic splitting conjecture. 55Q45, 55P60