Beta families arising from a v29 self-map on S∕(3,v18)

Belmont, Eva; Shimomura, Katsumi

doi:10.2140/agt.2023.23.4349

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Beta families arising from a $v_2^9$ self-map on $S/(3,v_1^8)$

Eva Belmont and Katsumi Shimomura

Algebraic & Geometric Topology 23 (2023) 4349–4378

DOI: 10.2140/agt.2023.23.4349

Abstract

We show that $v_{2}^{9}$ is a permanent cycle in the $3$ –primary Adams–Novikov spectral sequence computing $π_{*} (S ∕ (3, v_{1}^{8}))$ , and use this to conclude that the families $β_{9 t + 3 ∕ i}$ for $i = 1, 2$ , $β_{9 t + 6 ∕ i}$ for $i = 1, 2, 3$ , $β_{9 t + 9 ∕ i}$ for $i = 1, \dots, 8$ , $α_{1} β_{9 t + 3 ∕ 3}$ , and $α_{1} β_{9 t + 7}$ are permanent cycles in the $3$ –primary Adams–Novikov spectral sequence for the sphere for all $t \geq 0$ . We use a computer program by Wang to determine the additive and partial multiplicative structure of the Adams–Novikov $E_{2}$ page for the sphere in relevant degrees. The $i = 1$ cases recover previously known results of Behrens and Pemmaraju and the second author. The results about $β_{9 t + 3 ∕ 3}$ , $β_{9 t + 6 ∕ 3}$ and $β_{9 t + 9 ∕ 8}$ were previously claimed by the second author; the computer calculations allow us to give a more direct proof. As an application, we determine the image of the Hurewicz map $π_{*} S \to π_{*} tmf$ at $p = 3$ .

Keywords

Adams–Novikov spectral sequence, tmf, $v_2$–periodicity

Mathematical Subject Classification

Primary: 55Q45, 55Q51, 55T25

References

Publication

Received: 3 September 2021

Revised: 25 June 2022

Accepted: 23 July 2022

Published: 23 November 2023

Authors

Eva Belmont
Department of Mathematics, Applied Mathematics, and Statistics Case Western Reserve University Cleveland, OH United States

Katsumi Shimomura
Department of Mathematics, Faculty of Science Kochi University Kochi Japan

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Volume 23, issue 9 (2023)