A synthetic approach to detecting $v_1$-periodic families
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914406391087104 |
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| author | Carrick, Christian Davies, Jack Morgan |
| author_facet | Carrick, Christian Davies, Jack Morgan |
| contents | We provide a simple proof that the unit map from the sphere spectrum to the connective image-of-$J$ spectrum $\mathrm{j}$ is surjective on homotopy groups. This is achieved using a novel $t$-structure on the category of $E$-synthetic spectra and a specific construction of $\mathbf{F}_p$- and BP-synthetic lifts of $\mathrm{j}$. These synthetic lifts then easily produce modified Adams and Adams--Novikov spectral sequences for $\mathrm{j}$ which we use the prove the above detection statement, all without ever calculating $\mathbf{F}_p$- or BP-homology nor the associated Ext groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_16508 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A synthetic approach to detecting $v_1$-periodic families Carrick, Christian Davies, Jack Morgan Algebraic Topology K-Theory and Homology 55T05, 55T15, 55T25, 55N20, 55P42 We provide a simple proof that the unit map from the sphere spectrum to the connective image-of-$J$ spectrum $\mathrm{j}$ is surjective on homotopy groups. This is achieved using a novel $t$-structure on the category of $E$-synthetic spectra and a specific construction of $\mathbf{F}_p$- and BP-synthetic lifts of $\mathrm{j}$. These synthetic lifts then easily produce modified Adams and Adams--Novikov spectral sequences for $\mathrm{j}$ which we use the prove the above detection statement, all without ever calculating $\mathbf{F}_p$- or BP-homology nor the associated Ext groups. |
| title | A synthetic approach to detecting $v_1$-periodic families |
| topic | Algebraic Topology K-Theory and Homology 55T05, 55T15, 55T25, 55N20, 55P42 |
| url | https://arxiv.org/abs/2401.16508 |