Adams spectral sequences for non-vector-bundle Thom spectra
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911138552217600 |
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| author | Debray, Arun Yu, Matthew |
| author_facet | Debray, Arun Yu, Matthew |
| contents | When $R$ is one of the spectra $\mathit{ku}$, $\mathit{ko}$, $\mathit{tmf}$, $\mathit{MTSpin}^c$, $\mathit{MTSpin}$, or $\mathit{MTString}$, there is a standard approach to computing twisted $R$-homology groups of a space $X$ with the Adams spectral sequence, by using a change-of-rings isomorphism to simplify the $E_2$-page. This approach requires the assumption that the twist comes from a vector bundle, i.e. the twist map $X\to B\mathrm{GL}_1(R)$ factors through $B\mathrm{O}$. We show this assumption is unnecessary by working with Baker-Lazarev's Adams spectral sequence of $R$-modules and computing its $E_2$-page for a large class of twists of these spectra. We then work through two example computations motivated by anomaly cancellation for supergravity theories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_01678 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Adams spectral sequences for non-vector-bundle Thom spectra Debray, Arun Yu, Matthew Algebraic Topology High Energy Physics - Theory When $R$ is one of the spectra $\mathit{ku}$, $\mathit{ko}$, $\mathit{tmf}$, $\mathit{MTSpin}^c$, $\mathit{MTSpin}$, or $\mathit{MTString}$, there is a standard approach to computing twisted $R$-homology groups of a space $X$ with the Adams spectral sequence, by using a change-of-rings isomorphism to simplify the $E_2$-page. This approach requires the assumption that the twist comes from a vector bundle, i.e. the twist map $X\to B\mathrm{GL}_1(R)$ factors through $B\mathrm{O}$. We show this assumption is unnecessary by working with Baker-Lazarev's Adams spectral sequence of $R$-modules and computing its $E_2$-page for a large class of twists of these spectra. We then work through two example computations motivated by anomaly cancellation for supergravity theories. |
| title | Adams spectral sequences for non-vector-bundle Thom spectra |
| topic | Algebraic Topology High Energy Physics - Theory |
| url | https://arxiv.org/abs/2305.01678 |