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A Toda Bracket Convergence Theorem for Multiplicative Spectral Sequences

https://doi.org/10.1007/s42543-025-00099-x

Belmont, Eva; Kong, Hana_Jia (June 2025, Peking Mathematical Journal)

Abstract Moss’ theorem, which relates Massey products in the$$E_r$$ $E_{r}$ -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.
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ℝ-motivic v1-periodic homotopy

https://doi.org/10.2140/pjm.2024.330.43

Belmont, Eva; Isaksen, Daniel C; Kong, Hana Jia (July 2024, Pacific Journal of Mathematics)

We compute the $$v_1$$-periodic $$\mathbb{R}$$-motivic stable homotopy groups. The main tool is the effective slice spectral sequence. Along the way, we also analyze $$\mathbb{C}$$-motivic and $$\eta$$-periodic $$v_1$$-periodic homotopy from the same perspective.
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The reduced ring of the 𝑅𝑂(𝐶₂)-graded 𝐶₂-equivariant stable stems

https://doi.org/10.1090/bproc/203

Belmont, Eva; Xu, Zhouli; Zhang, Shangjie (January 2024, Proceedings of the American Mathematical Society, Series B)

We describe in terms of generators and relations the ring structure of the $R O (C_{2})$ -graded $C_{2}$ -equivariant stable stems ${π<#comment/>}_{⋆<#comment/>}^{C_{2}}$ modulo the ideal of all nilpotent elements. As a consequence, we also record the ring structure of the homotopy groups of the rational $C_{2}$ -equivariant sphere ${π<#comment/>}_{⋆<#comment/>}^{C_{2}} (S_{Q})$ .
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