In this paper we develop techniques to compute the cooperations algebra for the second truncated Brown-Peterson spectrum BP 2 . We prove that the cooperations algebra BP 2 * BP 2 decomposes as a direct sum of a F 2 -vector space concentrated in Adams filtration 0 and a F 2 [v 0 , v 1 , v 2 ]-module which is concentrated in even degrees and is v 2 -torsion free. We also develop a recursive method which produces a basis for the v 2 -torsion free part.There are two main parts of this paper. The first is a structural result regarding the algebra BP 2 * BP 2 . In particular, we will show there is a direct sum decomposition into a vector space V concentrated in Adams filtration 0, and a v 2 -torsion free component. The second is an inductive calculation of BP 2 * BP 2 . This inductive calculation is similar to the one produced in [5]. Moreover, this decomposition of BP 2 * BP 2 implies that the methods developed in [3] to calculate the bo-ASS can be applied to the BP 2 -ASS. One of our goals for later work is to prove an analogous splitting for tmf * tmf and develop the tmf-resolution as a computational device.Conventions. We will let A denote the mod 2 Steenrod algebra and A * its dual. We will let ζ k denote the conjugate of the the generator ξ k in the dual Steenrod algebra A * . We will also make the convention that ζ 0 = 1. Given a Hopf algebra B and a comodule M over B, we will often abbreviate Ext B (F 2 , M ) to Ext B (M ). Homology and cohomology are implicitly with mod 2 coefficients. All spectra are implicitly 2-complete.Also, we will use the notation E(n) to denote the subalgebra of A generated by the Milnor primitives Q 0 , . . . , Q n . This is in conflict with the standard notation for the Johnson-Wilson theories, but as these never arise in this paper, this will not present an issue.

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.

The E1‐term of the (2‐local) normalbo‐based Adams spectral sequence for the sphere spectrum decomposes into a direct sum of a v1‐periodic part, and a v1‐torsion part. Lellmann and Mahowald completely computed the d1‐differential on the v1‐periodic part, and the corresponding contribution to the E2‐term. The v1‐torsion part is harder to handle, but with the aid of a computer it was computed through the 20‐stem by Davis. Such computer computations are limited by the exponential growth of v1‐torsion in the E1‐term. In this paper, we introduce a new method for computing the contribution of the v1‐torsion part to the E2‐term, whose input is the cohomology of the Steenrod algebra. We demonstrate the efficacy of our technique by computing the normalbo‐Adams spectral sequence beyond the 40‐stem.

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