The framed Pontryagin Thom construction produces a graded ring isomorphism from the framed cobordism ring $\Omega^{fr}_*$ to the stable ring of homotopy groups of spheres $\pi_*(\mathbb{S})$. I have a few questions about geometric interpretations of algebraic facts under this correspondence.
- Every framed manifold has vanishing Stiefel-Whitney numbers, which implies that it is unoriented nullbordant, i.e. the zero element in $\Omega_*^O$. Given a manifold and a framing, is there a way to abstractly construct an unoriented nullbordism?
- By Serre's finiteness, every element of positive degree in $\pi_*(\mathbb{S})$ is torsion. Given a framed manifold $M$, can we prove "geometrically" that there exists an integer $k$ such that the disjoint union of $k$ copies of $M$ is framed null?
- Similarly by Nishida's nilpotence, given a framed manifold $M$, can we prove "geometrically" that there exists an integer $k'$ such that the $k'$-fold product of $M$ is framed null?
- A priori, the two integers $k$ and $k'$ defined in the questions above are framed bordism invariants. But do they really depend on the framing? In other words, can we find a manifold $M$ and two "non-bordant" framings such that $M^{k'}$ (or $\sqcup^k M$) is null with one framing but not with the other?
I am aware that, at first glance, I am basically asking for geometric proofs of very involved results by Serre and Nishida, so interesting special cases and examples to any of the questions, such as in this thread, would suffice!
