RO(G)-Graded Homotopy Fixed Point Spectral Sequence for Height 2 Morava E-Theory

Abstract

We consider

G=Q_8,{\text{SD}}_{16},G_{24},

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$G=Q_8,\textrm{SD}_{16},G_{24},$$\end{document}

and

G_{48}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$G_{48}$$\end{document}

as finite subgroups of the Morava stabilizer group which acts on the height 2 Morava E-theory

{\mathbf{E}}_2

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\textbf{E}}_2$$\end{document}

at the prime 2. We completely compute the G-homotopy fixed point spectral sequences of

{\mathbf{E}}_2

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\textbf{E}}_2$$\end{document}

. Our computation uses recently developed equivariant techniques since Hill, Hopkins, and Ravenel. We also compute the

(\ast -{\sigma }_i)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(*-\sigma _i)$$\end{document}

-graded

Q_8

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$Q_8$$\end{document}

- and

{\text{SD}}_{16}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textrm{SD}_{16}$$\end{document}

-homotopy fixed point spectral sequences, where

{\sigma }_i

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma _i$$\end{document}

is a non-trivial one-dimensional representation of

Q_8

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$Q_8$$\end{document}

.

Keywords

Morava E-theory / Topological modular forms / RO(G)-graded homotopy groups / 55P42 / 20J06 / 55Q91 / 55P60

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Funding

Shanghai Science and Technology Development Foundation(No. 20QA1401600)

National Science Foundation(No. DMS-1926686)

RIGHTS & PERMISSIONS

Peking University