A Toda Bracket Convergence Theorem for Multiplicative Spectral Sequences

Abstract

Moss’ theorem, which relates Massey products in the

E_r

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

-page of the classical Adams spectral sequence to Toda brackets of homotopy groups, is one of the main tools for calculating Adams differentials. Working in an arbitrary symmetric monoidal stable simplicial model category, we prove a general version of Moss’ theorem which applies to spectral sequences that arise from filtrations compatible with the monoidal structure. This involves the study of Massey products and Toda brackets in a non-strictly associative context. The theorem has broad applications, e.g., to the computation of the motivic slice spectral sequence and other colocalization towers.

Keywords

Toda bracket / Massey product / Spectral sequence / Motivic homotopy theory / Slice spectral sequence

References

[1] Burklund, R.: Multiplicative structures on Moore spectra. arXiv:2203.14787 (2022)

[2] ChristensenJD, FranklandM. Higher Toda brackets and the Adams spectral sequence in triangulated categories. Algebr. Geom. Topol., 2017, 1752687-2735.

[3] DavisDM, SnaithVP. Massey products and H(p,q)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H(p,\, q)$$\end{document}-systems. Math. Z., 1973, 132: 1-10.

[4] DuggerD, IsaksenDC. Motivic Hopf elements and relations. New York J. Math., 2013, 19: 823-871

[5] GheorgheB, IsaksenDC, KrauseA, RickaN. 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, 24103597-3628.

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

[7] GoerssPG, JardineJFSimplicial Homotopy. Theory Progress in Mathematics, 1999BaselBirkhäuser Verlag. 174

[8] GutiérrezJJ, RöndigsO, SpitzweckM, ØstværPA. Motivic slices and coloured operads. J. Topol., 2012, 53727-755.

[9] Hedenlund, A., Rognes, J.: A multiplicative Tate spectral sequence for compact Lie group actions. Mem. Amer. Math. Soc. 294(1468), v+134 pp. (2024)

[10] Hu, P., Kriz, I., Ormsby, K.: Convergence of the motivic Adams spectral sequence. J. K-theory 7(3), 573–596 (2011)

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

[12] IsaksenDC, WangG, XuZ. Stable homotopy groups of spheres: from dimension 0 to 90. Publ. Math. IHES, 2023, 137: 107-243.

[13] KochmanSO. Uniqueness of Massey products on the stable homotopy of spheres. Can. J. Math., 1980, 323576-589.

[14] Kochman, S.O.: Stable Homotopy Groups of Spheres—A Computer-Assisted Approach. Lecture Notes in Mathematics, vol. 1423. Springer, Berlin (1990)

[15] LevineM. Convergence of Voevodsky’s slice tower. Doc. Math., 2013, 18: 907-941.

[16] Massey, W.S.: Some higher order cohomology operations. In: Symposium Internacional de Topología Algebraica International Symposium on Algebraic Topology, pp. 145–154. Universidad Nacional Autónoma de México and UNESCO, Mexico City (1958)

[17] MayJP. Matric Massey products. J. Algebra, 1969, 12: 533-568.

[18] McCleary, J.: A User’s Guide to Spectral Sequences, 2nd edn. Cambridge Studies in Advanced Mathematics, vol. 58. Cambridge University Press, Cambridge (2001)

[19] Morel, F.: On the motivic π0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\pi _0$$\end{document} of the sphere spectrum. In: Axiomatic. Enriched and Motivic Homotopy Theory, pp. 219–260. Kluwer Academic Publishers, Dordrecht (2004)

[20] MossRMF. Secondary compositions and the Adams spectral sequence. Math. Z., 1970, 115: 283-310.

[21] Pelaez, P.: Multiplicative Properties of the Slice Filtration. Astérisque 335, xvi+289 pp. (2011)

[22] PstrągowskiP. Synthetic spectra and the cellular motivic category. Invent. Math., 2023, 2322553-681.

[23] RavenelDCComplex Cobordism and Stable Homotopy Groups of Spheres. Pure and Applied Mathematics, 1986Orlando, FLAcademic Press121

[24] Robinson, A.: Obstruction theory and the strict associativity of Morava K\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$K$$\end{document}-theories. In: Advances in Homotopy Theory (Cortona, 1988), London Math. Soc. Lecture Note Ser., vol. 139, pp. 143–152. Cambridge University Press, Cambridge (1989)

[25] RöndigsO, SpitzweckM, ØstværPA. The first stable homotopy groups of motivic spheres. Ann. Math. (2), 2019, 18911-74.

[26] Toda, H.: Composition Methods in Homotopy Groups of Spheres. Annals of Mathematics Studies, no. 49. Princeton University Press, Princeton, NJ (1962)

[27] Ullman, J.R.: On the regular slice spectral sequence. Ph.D. Thesis, Massachusetts Institute of Technology (2013)

Funding

National Science Foundation(DMS-2204357)

RIGHTS & PERMISSIONS

The Author(s)

Just Accepted

This article has successfully passed peer review and final editorial review, and will soon enter typesetting, proofreading and other publishing processes. The currently displayed version is the accepted final manuscript. The officially published version will be updated with format, DOI and citation information upon launch. We recommend that you pay attention to subsequent journal notifications and preferentially cite the officially published version. Thank you for your support and cooperation.