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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
Abstract

We show that v29 is a permanent cycle in the 3–primary Adams–Novikov spectral sequence computing π(S(3,v18)), and use this to conclude that the families β9t+3i for i = 1,2, β9t+6i for i = 1,2,3, β9t+9i for i = 1,,8, α1β9t+33, and α1β9t+7 are permanent cycles in the 3–primary Adams–Novikov spectral sequence for the sphere for all t 0. We use a computer program by Wang to determine the additive and partial multiplicative structure of the Adams–Novikov E2 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 β9t+33, β9t+63 and β9t+98 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 πS πtmf 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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