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$\mathrm{H}\mathbb{F}_2$-synthetic homotopy groups of topological modular forms

Peter Marek

Abstract

To any Adams-type spectrum $E$, Pstrągowski produced a symmetric monoidal stable $\infty$-category $Syn_E$ whose objects are, in a sense, ''formal Adams spectral sequences''. $Syn_E$ comes equipped with a lax symmetric monoidal functor $ν_E:Sp\to Syn_E$ from classical spectra, which embeds $Sp$ fully and faithfully in $Syn_E$, and is a category with a natural notion of bigraded homotopy groups. The bigraded homotopy groups $π_{*,*}ν_EX$ systematically record information about the homotopy groups $π_*X$ and the $E$-Adams spectral sequence of $X$. In this paper, we compute the $ν_{\mathrm{H}\mathbb{F}_2}\mathbb{F}_2$-Adams spectral sequence of $ν_{\mathrm{H}\mathbb{F}_2}tmf_2^{\wedge}$, synthetic versions of hidden $2$-, $η$-, $ν$-, and $\overlineκ$-extensions, and use this to deduce information about the homotopy ring structure of $π_{*,*}ν_{\mathrm{H}\mathbb{F}_2}tmf_2^{\wedge}$.

$\mathrm{H}\mathbb{F}_2$-synthetic homotopy groups of topological modular forms

Abstract

To any Adams-type spectrum , Pstrągowski produced a symmetric monoidal stable -category whose objects are, in a sense, ''formal Adams spectral sequences''. comes equipped with a lax symmetric monoidal functor from classical spectra, which embeds fully and faithfully in , and is a category with a natural notion of bigraded homotopy groups. The bigraded homotopy groups systematically record information about the homotopy groups and the -Adams spectral sequence of . In this paper, we compute the -Adams spectral sequence of , synthetic versions of hidden -, -, -, and -extensions, and use this to deduce information about the homotopy ring structure of .
Paper Structure (16 sections, 91 theorems, 72 equations, 8 figures, 6 tables)

This paper contains 16 sections, 91 theorems, 72 equations, 8 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Overview of Synthetic Spectra
  4. Synthetic Adams Spectral Sequence
  5. E2-page and Ri'-modules
  6. E3-page
  7. E4-page
  8. E-infinity page
  9. Synthetic Multiplicative Structure
  10. Synthetic Hidden Extensions
  11. Synthetic Hidden Extension Proofs
  12. Tilde 2-hidden extensions
  13. Eta-hidden extensions
  14. nu-hidden extensions
  15. Other extensions and relations
  16. ...and 1 more sections

Key Result

Theorem 1.1

(Section AdamsSScalcdetailssection) The ${}_{\mathrm{syn}}\mathrm{E}_r^{*,*,*}$-pages of the $\nu{\mathrm{H}}{\mathbb{F}}_2$-Adams spectral sequence for $\nu_{{\mathrm{H}}{\mathbb{F}}_2}tmf$ are completely computed. The $R_0'$-module structure of ${}_{\mathrm{syn}}\mathrm{E}_2^{*,*,*}$, the $R_1'$-m

Figures (8)

  • Figure 1: $\pi_{k,w}(tmf)$, $0\leq k\leq 24$
  • Figure 2: $\pi_{k,w}(tmf)$, $24\leq k\leq 48$
  • Figure 3: $\pi_{k,w}(tmf)$, $48\leq k\leq 72$
  • Figure 4: $\pi_{k,w}(tmf)$, $72\leq k\leq 96$
  • Figure 5: $\pi_{k,w}(tmf)$, $96\leq k\leq 120$
  • ...and 3 more figures

Theorems & Definitions (181)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.1
  • Corollary 2.2
  • proof
  • Definition 2.3
  • ...and 171 more