On products of beta and gamma elements in the homotpy of the first Smith-Toda spectrum

Katsumi Shimomura, Mao-No-Suke Shimomura

Publication date: 4 September 2022

Abstract: In this paper, we determine the first cohomology of the monochromatic comodule

M_{2}^{1}

at an odd prime, and apply the results to show non-trivialities 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 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.

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