On products of beta and gamma elements in the homotpy of the first Smith-Toda spectrum
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866917870519189504 |
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| author | Shimomura, Katsumi Shimomura, Mao-No-Suke |
| author_facet | Shimomura, Katsumi Shimomura, Mao-No-Suke |
| contents | In this paper, we determine the first cohomology of the monochromatic comodule $M^1_2$ 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 $M^3_0$, which we hope to determine for a long time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_01742 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On products of beta and gamma elements in the homotpy of the first Smith-Toda spectrum Shimomura, Katsumi Shimomura, Mao-No-Suke Algebraic Topology In this paper, we determine the first cohomology of the monochromatic comodule $M^1_2$ 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 $M^3_0$, which we hope to determine for a long time. |
| title | On products of beta and gamma elements in the homotpy of the first Smith-Toda spectrum |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2209.01742 |