The fold-map cobordism conjecture for powers of two

Let $n>k\geq 2$ and let $k=2^a-1$ for some $a\geq 2$. Let $M$ be an $n$-manifold, let $\mathfrak N_n$ denote the unoriented cobordism group in dimension $n$, and suppose that $w_i(TM)=0$ for $i=1,\ldots,k$. A fold map is a smooth map with only fold singularities; a stably parallelizable manifold is one whose stable tangent bundle is trivial.

Fold-map cobordism conjecture. There exists a $1$-dimensional linear subspace $\mathfrak G^1\leq\mathfrak N_n$ such that, if $M$ admits a fold map into a stably parallelizable $(n-k)$-manifold, then either

$$n=2^s\text{ or }n=2^s+1\quad\text{with }s\geq a+1,$$

and $[M]\in\mathfrak G^1$, or $M$ is null-cobordant.

This conjecture is motivated by numerical calculations analogous to those used in the proof of the preceding theorem. It predicts that, under the stated Stiefel–Whitney and fold-map hypotheses, the only potentially nontrivial cobordism classes occur in the indicated dimensions; its resolution is not supplied here.