Classical Stable Homotopy Groups of Spheres via

{\mathbb{F}}_2

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

-Synthetic Methods

Abstract

We study the

{\mathbb{F}}_2

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

-synthetic Adams spectral sequence. We obtain new computational information about

\mathbb{C}

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

-motivic and classical stable homotopy groups.

Keywords

Stable homotopy group / Synthetic homotopy theory / Adams spectral sequence / Primary 55Q45 / Secondary 55T15

References

[1] Adams, J.F.: On the non-existence of elements of Hopf invariant one. Ann. of Math. (2) 72, 20–104 (1960)

[2] Baer, J.F., Johnson, M., Marek, P.: Stable comodule deformations and the synthetic Adams–Novikov spectral sequence. arXiv:2402.14274 (2024)

[3] Barratt, M.G., Jones, J.D.S., Mahowald, M.E.: Relations amongst Toda brackets and the Kervaire invariant in dimension 62. J. London Math. Soc. (2) 30(3), 533–550 (1984)

[4] Belmont, E., Kong, H.J.: A Toda bracket convergence theorem for multiplicative spectral sequences. arXiv:2112.08689 (2021)

[5] Burklund R. An extension in the Adams spectral sequence in dimension 54. Bull. Lond. Math. Soc., 2021, 53(2): 404-407

[6] Burklund R, Hahn J, Senger A. On the boundaries of highly connected, almost closed manifolds. Acta Math., 2023, 231(2): 205-344

[7] Burklund R, Xu Z. The Adams differentials on the classes hj3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\text{h}{^{3}_{j}}$$\end{document}. Invent. Math., 2025, 239(1): 1-77

[8] Gheorghe B, Isaksen DC, Krause A, Ricka N. C\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{C}$$\end{document}-motivic modular forms. J. Eur. Math. Soc. (JEMS), 2022, 24(10): 3597-3628

[9] Gheorghe B, Wang G, Xu Z. The special fiber of the motivic deformation of the stable homotopy category is algebraic. Acta Math., 2021, 226(2): 319-407

[10] Hopkins MJ , Lin J, Shi XD, Xu Z. Intersection forms of spin 4-manifolds and the Pin(2)-equivariant Mahowald invariant. Comm. Am. Math. Soc., 2022, 2: 22-132

[11] Isaksen, D.C.: Stable stems. Mem. Am. Math. Soc. 262(1269), viii+159 pp. (2019)

[12] Isaksen, D.C., Wang, G., Xu, Z.: Classical and C\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{C}$$\end{document}-motivic Adams charts. https://cpb-us-e1.wpmucdn.com/s.wayne.edu/dist/0/60/files/2020/01/Adamscharts.pdf (2020)

[13] Isaksen DC , Wang G, Xu Z. Stable homotopy groups of spheres. Proc. Natl. Acad. Sci. U.S.A., 2020, 117(40): 24757-24763

[14] Isaksen DC, Wang G, Xu Z. Stable homotopy groups of spheres: from dimension 0 to 90. Publ. Math. Inst. Hautes Études Sci., 2023, 137: 107-243

[15] Lin, W., Wang, G., Xu, Z.: Machine proofs for Adams differentials and extension problems among CW spectra. arXiv:2412.10876 (2024)

[16] Marek, P.: HF2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{F}_2$$\end{document}-synthetic homotopy groups of topological modular forms. arXiv:2202.11305 (2022)

[17] Moss RMF . Secondary compositions and the Adams spectral sequence. Math. Z., 1970, 115: 283-310

[18] Pstragowski P. Synthetic spectra and the cellular motivic category. Invent. Math., 2023, 232(2): 553-681

[19] Serre J-P. Groupes d’homotopie et classes de groupes abéliens. Ann. of Math. (2), 1953, 58: 258-294

[20] Wang G, Xu Z. The triviality of the 61-stem in the stable homotopy groups of spheres. Ann. of Math. (2), 2017, 186(2): 501-580

[21] Xu Z. The strong Kervaire invariant problem in dimension 62. Geom. Topol., 2016, 20(3): 1611-1624

Funding

National Science Foundation(DMS-2202992)

Villum Fonden(Young Investigator Program)

Danish National Research Foundation(Copenhagen center for Geometry and Topology (DNRF151))

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