$\mathrm{H}\mathbb{F}_2$-synthetic homotopy groups of topological modular forms
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909115492597760 |
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| author | Marek, Peter |
| author_facet | Marek, Peter |
| contents | To any Adams-type spectrum $E$, Pstrągowski produced a symmetric monoidal stable $\infty$-category $Syn_E$ whose objects are, in a sense, ''formal Adams spectral sequences''. $Syn_E$ comes equipped with a lax symmetric monoidal functor $ν_E:Sp\to Syn_E$ from classical spectra, which embeds $Sp$ fully and faithfully in $Syn_E$, and is a category with a natural notion of bigraded homotopy groups. The bigraded homotopy groups $π_{*,*}ν_EX$ systematically record information about the homotopy groups $π_*X$ and the $E$-Adams spectral sequence of $X$. In this paper, we compute the $ν_{\mathrm{H}\mathbb{F}_2}\mathbb{F}_2$-Adams spectral sequence of $ν_{\mathrm{H}\mathbb{F}_2}tmf_2^{\wedge}$, synthetic versions of hidden $2$-, $η$-, $ν$-, and $\overlineκ$-extensions, and use this to deduce information about the homotopy ring structure of $π_{*,*}ν_{\mathrm{H}\mathbb{F}_2}tmf_2^{\wedge}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_11305 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | $\mathrm{H}\mathbb{F}_2$-synthetic homotopy groups of topological modular forms Marek, Peter Algebraic Topology 55Q10 To any Adams-type spectrum $E$, Pstrągowski produced a symmetric monoidal stable $\infty$-category $Syn_E$ whose objects are, in a sense, ''formal Adams spectral sequences''. $Syn_E$ comes equipped with a lax symmetric monoidal functor $ν_E:Sp\to Syn_E$ from classical spectra, which embeds $Sp$ fully and faithfully in $Syn_E$, and is a category with a natural notion of bigraded homotopy groups. The bigraded homotopy groups $π_{*,*}ν_EX$ systematically record information about the homotopy groups $π_*X$ and the $E$-Adams spectral sequence of $X$. In this paper, we compute the $ν_{\mathrm{H}\mathbb{F}_2}\mathbb{F}_2$-Adams spectral sequence of $ν_{\mathrm{H}\mathbb{F}_2}tmf_2^{\wedge}$, synthetic versions of hidden $2$-, $η$-, $ν$-, and $\overlineκ$-extensions, and use this to deduce information about the homotopy ring structure of $π_{*,*}ν_{\mathrm{H}\mathbb{F}_2}tmf_2^{\wedge}$. |
| title | $\mathrm{H}\mathbb{F}_2$-synthetic homotopy groups of topological modular forms |
| topic | Algebraic Topology 55Q10 |
| url | https://arxiv.org/abs/2202.11305 |