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.


1. 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?
2. 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?
3. 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?
4. 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](https://mathoverflow.net/questions/201587/nilpotence-of-the-stable-hopf-map-via-framed-cobordism), would suffice!



Torsion and nilpotence of framed manifolds under the Pontryagin-Thom map
1. I don't know how satisfied you would be by the paper:


"AN ELEMENTARY GEOMETRIC PROOF OF TWO THEOREMS OF THOM"
SANDRO BUONCRISTIANO and DEREK HACON


4. I think this happens already with the framed circle. One framing is nullbordant, but the other one is not (so k=1 or k=2). But maybe I don't understand the question.


