Higher chromatic Thom spectra via unstable homotopy theory

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 $\mathrm{M}\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}\to \mathrm{k}\mathrm{o}$ and $\mathrm{M}\mathrm{S}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}\to \mathrm{t}\mathrm{m}\mathrm{f}$ 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 $\mathrm{B}\mathrm{P}⟨n-1⟩$, $\mathrm{k}\mathrm{o}$, and $\mathrm{t}\mathrm{m}\mathrm{f}$ as Thom spectra (albeit over $T(n)$, $A$, and $B$ respectively, and not over the sphere). This interpretation of $\mathrm{B}\mathrm{P}⟨n-1⟩$, $\mathrm{k}\mathrm{o}$, and $\mathrm{t}\mathrm{m}\mathrm{f}$ offers a new perspective on Wood equivalences of the form $\mathrm{b}\mathrm{o}\wedge C\eta \simeq \mathrm{b}\mathrm{u}$: they are related to the existence of certain EHP sequences in unstable homotopy theory. This construction of $\mathrm{B}\mathrm{P}⟨n-1⟩$ also provides a different lens on the nilpotence theorem. Finally, we prove a $C_2$-equivariant analogue of our construction, describing $\underset{‾}{\mathrm{H}\mathbf{Z}}$ as a Thom spectrum.