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.
Categories: math.AT