Higher chromatic Thom spectra via unstable homotopy theory
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866917622206955520 |
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| author | Devalapurkar, Sanath K |
| author_facet | Devalapurkar, Sanath K |
| contents | We investigate implications of an old conjecture in unstable homotopy theory related to the Cohen-Moore-Neisendorfer theorem and a conjecture about the $\mathbf{E}_{2}$-topological Hochschild cohomology of certain Thom spectra (denoted $A$, $B$, and $T(n)$) related to Ravenel's $X(p^n)$. We show that these conjectures imply that the orientations $\mathrm{MSpin}\to \mathrm{ko}$ and $\mathrm{MString}\to \mathrm{tmf}$ admit spectrum-level splittings. This is shown by generalizing a theorem of Hopkins and Mahowald, which constructs $\mathrm{H}\mathbf{F}_p$ as a Thom spectrum, to construct $\mathrm{BP}\langle{n-1}\rangle$, $\mathrm{ko}$, and $\mathrm{tmf}$ as Thom spectra (albeit over $T(n)$, $A$, and $B$ respectively, and not over the sphere). This interpretation of $\mathrm{BP}\langle{n-1}\rangle$, $\mathrm{ko}$, and $\mathrm{tmf}$ offers a new perspective on Wood equivalences of the form $\mathrm{bo} \wedge Cη\simeq \mathrm{bu}$: they are related to the existence of certain EHP sequences in unstable homotopy theory. This construction of $\mathrm{BP}\langle{n-1}\rangle$ also provides a different lens on the nilpotence theorem. Finally, we prove a $C_2$-equivariant analogue of our construction, describing $\underline{\mathrm{H}\mathbf{Z}}$ as a Thom spectrum. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2004_08951 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Higher chromatic Thom spectra via unstable homotopy theory Devalapurkar, Sanath K Algebraic Topology We investigate implications of an old conjecture in unstable homotopy theory related to the Cohen-Moore-Neisendorfer theorem and a conjecture about the $\mathbf{E}_{2}$-topological Hochschild cohomology of certain Thom spectra (denoted $A$, $B$, and $T(n)$) related to Ravenel's $X(p^n)$. We show that these conjectures imply that the orientations $\mathrm{MSpin}\to \mathrm{ko}$ and $\mathrm{MString}\to \mathrm{tmf}$ admit spectrum-level splittings. This is shown by generalizing a theorem of Hopkins and Mahowald, which constructs $\mathrm{H}\mathbf{F}_p$ as a Thom spectrum, to construct $\mathrm{BP}\langle{n-1}\rangle$, $\mathrm{ko}$, and $\mathrm{tmf}$ as Thom spectra (albeit over $T(n)$, $A$, and $B$ respectively, and not over the sphere). This interpretation of $\mathrm{BP}\langle{n-1}\rangle$, $\mathrm{ko}$, and $\mathrm{tmf}$ offers a new perspective on Wood equivalences of the form $\mathrm{bo} \wedge Cη\simeq \mathrm{bu}$: they are related to the existence of certain EHP sequences in unstable homotopy theory. This construction of $\mathrm{BP}\langle{n-1}\rangle$ also provides a different lens on the nilpotence theorem. Finally, we prove a $C_2$-equivariant analogue of our construction, describing $\underline{\mathrm{H}\mathbf{Z}}$ as a Thom spectrum. |
| title | Higher chromatic Thom spectra via unstable homotopy theory |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2004.08951 |