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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.17149 |
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| _version_ | 1866911593292365824 |
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| author | Li, Guchuan Petersen, Sarah Tatum, Elizabeth |
| author_facet | Li, Guchuan Petersen, Sarah Tatum, Elizabeth |
| contents | In the 1980's, Mahowald and Kane used integral Brown--Gitler spectra to decompose $ku \wedge ku$ as a sum of finitely generated $ku$-module spectra. This splitting, along with an analogous decomposition of $ko \wedge ko,$ led to a great deal of progress in stable homotopy computations and understanding of $v_1$-periodicity in the stable homotopy groups of spheres. In this paper, we construct a $C_2$-equivariant lift of Mahowald and Kane's splitting of $ku \wedge ku$. We also describe the resulting $C_2$-equivariant splitting in terms of $C_2$-equivariant Adams covers and record an analogous splitting for $H\underline{\mathbb{Z}} \wedge H \underline{\mathbb{Z}}$. Along the way, we give complete computations of the $ku_{\mathbb{R}}$ and $H \mathbb{Z}$ operations and cooperations algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_17149 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A spectrum-level splitting of the $ku_\mathbb{R}$-cooperations algebra Li, Guchuan Petersen, Sarah Tatum, Elizabeth Algebraic Topology In the 1980's, Mahowald and Kane used integral Brown--Gitler spectra to decompose $ku \wedge ku$ as a sum of finitely generated $ku$-module spectra. This splitting, along with an analogous decomposition of $ko \wedge ko,$ led to a great deal of progress in stable homotopy computations and understanding of $v_1$-periodicity in the stable homotopy groups of spheres. In this paper, we construct a $C_2$-equivariant lift of Mahowald and Kane's splitting of $ku \wedge ku$. We also describe the resulting $C_2$-equivariant splitting in terms of $C_2$-equivariant Adams covers and record an analogous splitting for $H\underline{\mathbb{Z}} \wedge H \underline{\mathbb{Z}}$. Along the way, we give complete computations of the $ku_{\mathbb{R}}$ and $H \mathbb{Z}$ operations and cooperations algebras. |
| title | A spectrum-level splitting of the $ku_\mathbb{R}$-cooperations algebra |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2503.17149 |