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on Pontryagin’s identification of unstable Cohomotopy of closed manifolds with cobordism classes of their normally framed submanifolds – to go alongside Thom’s theorem (which is originally the analogous statement but for oriented submanifolds and maps into a universal $SO(n)$-Thom space) and the Pontryagin-Thom construction (which has come to be the term used for all kinds of generalizations and variants that neither Pontryagin nor Thom probably ever dreamed of).
For the moment the bulk of the material is copied over from the existing section at Cohomotopy, but I hope to improve a bit on this a little later.
Here is a puzzlement I have about attribution:
Likely the most common version of the statement assumes that the manifold $M^d$ is closed (hence compact) and then finds the bijection of the form
$Cob^n_{Fr}(M^d) \;\simeq\; \pi_0 Maps \big( M^d, S^n \big) = \pi^n(M^d) \,.$For instance, this is what Kosinski states in his chapter IX, Prop. 5.5.
But since the left hand side here (cobordism classes of normally framed submanifolds) does not really need/want the compactness assumption, a more pleasant version of the statement would be
$Cob^n_{Fr}(M^d) \;\simeq\; \pi_0 Maps^{\ast/} \Big( \big( M^d \big)^{cpt} , S^n \Big) = \tilde \pi^n \Big( \big( M^d \big)^{cpt} \Big) \,,$where now on the right we have pointed maps out of the one-point compactification of $M^d$, naturally thought of as its reduced Cohomotopy.
This second version follows along the same lines as the first, and maybe many people may have the second version in mind when they think of the theorem. But for purposes of citation, to which source could the second version be attributed?
Looking around, I see the following two opposite extremes:
1) On p. 1 of the thesis “Homology stability for spaces of surfaces” (pdf) it’s attributed to the original article by Pontryagin from 1938 .
Now, while it’s plausible that Pontryagin would have readily provided the proof of this version of the theorem if asked for it, it is a stretch to claim that it’s in his 1938 article. Even Kosinski’s form of Pontryagin’s theorem is never really stated in Pontryagin’s articles.
2) In contrast, just last year somebody found it necessary to write out and publish a proof of the second version of the theorem above: It’s the last paragraphs of Csépai 20, proven there as part of a proof of a more general statement, but occupying a sizeable fraction of the whole article nonetheless.
Probably all this means that the second version has been obvious to anyone who thought about it, but that it was never ever brought to paper properly. Until last year!?
If anyone has another citable reference, please let me know.
I have expanded/re-written the Idea-section. Among other things, I have added a somewhat improved graphics illustrating the construction.
Have removed the previous “Details”-section for the moment, which was copied over from Cohomotopy (where it’s still present), since now notation would need to be harmonized with the new Idea-section. Hope to get to it later.
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