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 $\mathrm{tmf}$-resolutions. To this end, we compute the structure of the $\mathrm{tmf}$-resolution for a specific type 2 complex $Z$. We find that, analogous to the height 1 case, the $E_1$-page of the $\mathrm{tmf}$-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 $\mathrm{tmf}$-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 $\mathrm{tmf}$-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.

中文翻译：

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2高处的望远镜猜想和tmf分辨率

Mahowald 在素数 2 证明了高度 1 望远镜猜想，作为他对 bo 分辨率的开创性工作的应用。在本文中，我们通过透镜研究了素数 2 处的高 2 望远镜猜想$\mathrm{肌动蛋白}$-决议。为此，我们计算结构$\mathrm{肌动蛋白}$- 特定类型 2 复合体的分辨率$Z$. 我们发现，类似于高度 1 的情况，${乙}_1$- 页面$\mathrm{肌动蛋白}$-分辨率具有分解为$v_2$-periodic summand 和 Eilenberg–MacLane summand，它由有界$v_2$-扭转。然而，与高度 1 的情况不同，${乙}_2$- 页面$\mathrm{肌动蛋白}$-分辨率展品无界$v_2$-扭转。我们将其与 Mahowald-Ravenel-Shick 的工作进行比较，并讨论望远镜猜想的有效性如何与这个无界的命运联系起来$v_2$-torsion：无界$v_2$-torsion 在光谱序列中自行消失，望远镜猜想是正确的，或者它持续形成$v_2$-抛物线和望远镜猜想是错误的。我们还研究如何使用$\mathrm{肌动蛋白}$-分辨率有效地给出同伦群的低维计算$Z$. 这些计算使我们能够证明第二作者和 Egger 的一个猜想：$乙(2)$-local Adams–Novikov 谱序列$Z$ 崩溃。