On products of beta and gamma elements in the homotopy of the first Smith–Toda spectrum

Shimomura, Katsumi; Shimomura, Mao-no-suke

doi:10.2140/agt.2024.24.3875

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On products of beta and gamma elements in the homotopy of the first Smith–Toda spectrum

Katsumi Shimomura and Mao-no-suke Shimomura

Algebraic & Geometric Topology 24 (2024) 3875–3896

DOI: 10.2140/agt.2024.24.3875

Abstract

We determine the first cohomology of the monochromatic comodule $M_{2}^{1}$ at an odd prime, and apply the results to show nontrivialities of some products of beta and gamma elements in the homotopy groups of the Smith–Toda spectrum $V (1)$ . The cohomology for a prime greater than three was previously determined by the first author. Here we verify them and determine the cohomology at the prime 3 by elementary calculation. The cohomology will be a stepping stone for computing the cohomology of the monochromatic comodule $M_{0}^{3}$ , which we hope to determine for a long time.

Keywords

Smith–Toda spectra, stable homotopy groups, Greek letter elements, monochromatic comodules

Mathematical Subject Classification

Primary: 55Q45

Secondary: 55Q51, 55T15

References

Publication

Received: 7 November 2022

Revised: 23 July 2023

Accepted: 5 November 2023

Published: 9 December 2024

Authors

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

Mao-no-suke Shimomura
Department of Mathematics, Faculty of Science and Technology Kochi University Kochi Japan

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Volume 24, issue 7 (2024)