Towards the $K(2)$-local homotopy groups of $Z$

Abstract

Recently we introduced a class $\tilde{\mathcal{Z}}$ of $2$-local finite spectra and showed that all spectra $Z\in \tilde{\mathcal{Z}}$ admit a $v_2$-self-map of periodicity $1$. The aim of this article is to compute the $K(2)$-local homotopy groups ${\pi }_{\ast }{L}_{K(2)}Z$ of all spectra $Z\in \tilde{\mathcal{Z}}$ using a homotopy fixed point spectral sequence, and we give an almost complete computation. The incompleteness lies in the fact that we are unable to eliminate one family of $d_3$-differentials and a few potential hidden extensions, though we conjecture that all these differentials and hidden extensions are trivial.

References in corpus (1)

• Topological resolutions in K(2)-local homotopy theory at the prime 2

Cited by in corpus (2)

• The telescope conjecture at height 2 and the tmf resolution
• The topological modular forms of $\mathbb{R}{P}^2$ and $\mathbb{R}{P}^2\wedge \mathbb{C}{P}^2$