Higher chromatic Thom spectra via unstable homotopy theory

Abstract

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 $\mathrm{bo} \wedge C\eta \simeq \mathrm{bu}$: 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.