The telescope conjecture at height 2 and the tmf resolution
The telescope conjecture at height 2 and the tmf resolution
复制标题
DOI：
10.1112/topo.12208
复制
发表时间：
2019-09
影响因子：
1.1
通讯作者：
A. Beaudry;M. Behrens;P. Bhattacharya;D. Culver;Zhouli Xu
中科院分区：
文献类型：
--
作者：
A. Beaudry;M. Behrens;P. Bhattacharya;D. Culver;Zhouli Xu
Mahowald在质数2时证明了高度为1的望远镜猜想，作为他在bo-resolutions上的开创性工作的应用。本文通过tmf-分辨率的透镜研究了素数为2的高度为2的望远镜猜想。为此，我们计算一个特定的2型复形Z的tmf-分解的结构。我们发现，类似于高度为1的情况，tmf-分解的E1-页具有分解为v2-周期和项，以及由有界v2-挠率组成的Eilenberg-MacLane和项。然而，与高度为1的情况不同，tmf-分辨率的E2-页表现出无界的v2-挠率。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种无界v2-挠率的命运联系起来：要么无界v2-挠率在谱序列中杀死自己，望远镜猜想是真的，或者它持续形成v2-抛物线，望远镜猜想是假的。我们还研究了如何使用tmf-分解来有效地给出Z的同伦群的低维计算。这些计算使我们能够证明第二作者和Egger的一个猜想：Z坍缩的E（2）-局部Adams-Novikov谱序列。
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.