Classical Stable Homotopy Groups of Spheres via $\mathbb {F}_2$-Synthetic Methods

Abstract

We study the $\mathbb {F}_2$-synthetic Adams spectral sequence. We obtain new computational information about $\mathbb {C}$-motivic and classical stable homotopy groups.

Keywords

Stable homotopy group / Synthetic homotopy theory / Adams spectral sequence

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 $\text{h}{^{3}_{j}}$. Invent. Math., 2025, 239(1):1-77

[8.] Gheorghe B, Isaksen DC, Krause A, Ricka N. $\mathbb{C}$-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 $\mathbb{C}$-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.: H$\mathbb{F}_2$-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 http://dx.doi.org/10.13039/100000001(DMS-2105462); Villum Fonden http://dx.doi.org/10.13039/100008398(Young Investigator Program); Danish National Research Foundation(Copenhagen center for Geometry and Topology (DNRF151))