The K-theory cochains of H-spaces and height 1 chromatic homotopy theory
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866912835367337984 |
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| author | van Nigtevecht, Sven |
| author_facet | van Nigtevecht, Sven |
| contents | Fix an odd prime $p$. Let $X$ be a pointed space whose $p$-completed K-theory $\mathrm{KU}_p^*(X)$ is an exterior algebra on a finite number of odd generators; examples include odd spheres and many H-spaces. We give a generators-and-relations description of the $\mathbf{E}_\infty$-$\mathrm{KU}_p$-algebra spectrum $\mathrm{KU}_p^{X_+}$ of $\mathrm{KU}_p$-cochains of $X$. To facilitate this construction, we describe a $\mathrm{K}(1)$-local analogue of the Tor spectral sequence for $\mathbf{E}_1$-ring spectra. Combined with previous work of Bousfield, this description of the cochains of $X$ recovers a result of Kjaer that the $v_1$-periodic homotopy type of $X$ can be modelled by these cochains. This then implies that the Goodwillie tower of the height 1 Bousfield-Kuhn functor converges for such $X$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_07311 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The K-theory cochains of H-spaces and height 1 chromatic homotopy theory van Nigtevecht, Sven Algebraic Topology Fix an odd prime $p$. Let $X$ be a pointed space whose $p$-completed K-theory $\mathrm{KU}_p^*(X)$ is an exterior algebra on a finite number of odd generators; examples include odd spheres and many H-spaces. We give a generators-and-relations description of the $\mathbf{E}_\infty$-$\mathrm{KU}_p$-algebra spectrum $\mathrm{KU}_p^{X_+}$ of $\mathrm{KU}_p$-cochains of $X$. To facilitate this construction, we describe a $\mathrm{K}(1)$-local analogue of the Tor spectral sequence for $\mathbf{E}_1$-ring spectra. Combined with previous work of Bousfield, this description of the cochains of $X$ recovers a result of Kjaer that the $v_1$-periodic homotopy type of $X$ can be modelled by these cochains. This then implies that the Goodwillie tower of the height 1 Bousfield-Kuhn functor converges for such $X$. |
| title | The K-theory cochains of H-spaces and height 1 chromatic homotopy theory |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2209.07311 |