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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+3∕i for i = 1,2, β9t+6∕i for i = 1,2,3, β9t+9∕i for i = 1,… ,8, α1β9t+3∕3, 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+3∕3, β9t+6∕3 and β9t+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 → π∗tmf at p = 3.
Belmont, Eva 1 ; Shimomura, Katsumi 2
@article{10_2140_agt_2023_23_4349,
author = {Belmont, Eva and Shimomura, Katsumi},
title = {Beta families arising from a v29 self-map on {S\ensuremath{/}(3,v18)}},
journal = {Algebraic and Geometric Topology},
pages = {4349--4378},
publisher = {mathdoc},
volume = {23},
number = {9},
year = {2023},
doi = {10.2140/agt.2023.23.4349},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2023.23.4349/}
}
TY - JOUR AU - Belmont, Eva AU - Shimomura, Katsumi TI - Beta families arising from a v29 self-map on S∕(3,v18) JO - Algebraic and Geometric Topology PY - 2023 SP - 4349 EP - 4378 VL - 23 IS - 9 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2023.23.4349/ DO - 10.2140/agt.2023.23.4349 ID - 10_2140_agt_2023_23_4349 ER -
%0 Journal Article %A Belmont, Eva %A Shimomura, Katsumi %T Beta families arising from a v29 self-map on S∕(3,v18) %J Algebraic and Geometric Topology %D 2023 %P 4349-4378 %V 23 %N 9 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2023.23.4349/ %R 10.2140/agt.2023.23.4349 %F 10_2140_agt_2023_23_4349
Belmont, Eva; Shimomura, Katsumi. Beta families arising from a v29 self-map on S∕(3,v18). Algebraic and Geometric Topology, Tome 23 (2023) no. 9, pp. 4349-4378. doi: 10.2140/agt.2023.23.4349
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