This is a slightly revamped version of a [question](https://math.stackexchange.com/questions/241159/suspension-operation-on-the-pontryagin-thom-construction) I asked on the stackexchange forum. That question was asking if the Pontryagin-Thom constructon respects the suspension operation, alluding to stable homotopy. I now would like to dig deeper on this allusion.


For a compact simply-connected oriented manifold $M$, view the Pontryagin-Thom construction as the bijective correspondence between $[M,S^r]$ and the set of (appropriate equivalence classes of) framed submanifolds of codimension $r$ in $M$. A quick subquestion is: **Is this in some way functorial?**  

For applying suspension $\Sigma^1$, I would ideally like to keep our general $M$, but for now I would be happy understanding what occurs for $M\approx S^n$. I am actually unsure if this will work for general $M$ because suspension doesn't typically produce a manifold right? -- Maybe we can tweak it (smooth the corners, or homotope it) to produce a manifold.  

**In perhaps a non-rigorous sense, does $\Sigma^1$ represent a natural transformation for the above construction?** In particular, under $\Sigma^1$ we pass from $[M,S^r]$ to $[\Sigma^1M,S^{r+1}]$. So I would expect to get a correspondence $\Sigma^1\lbrace\text{framed }(n-r)\text{-submanifolds in }M\rbrace\simeq\lbrace\text{framed }(n-r)\text{-submanifolds in }\Sigma^1M\rbrace$.


Now if this all works out, what happens on repeated iterations $\Sigma^n$? **Does the Pontryagin-Thom construction blossom into anything under the infinite-suspension $\Sigma^\infty$?**



(Infinite) Suspension Functor on the Pontryagin-Thom Construction
Johannes has answered your first two questions, let me have a stab at the final question.


The answer is yes, it blossoms into bordism classes of framed *immersions* of codimension $r$. That is to say, the pointed stable homotopy group 
$$\lim\_{n\to\infty}[\Sigma^n M\_+, S^{n+r}]\cong [M\_+,\Omega^\infty \Sigma^\infty S^r] $$
is in one-to-one correspondence with the bordism classes of framed immersions $N\looparrowright M$ of codimension $r$, at least when $r\geq 1$. (Here the $+$ denotes one-point compactification, which means adding a disjoint base-point since $M$ is already compact.) This is sometimes called the Pontrjagin-Thom-Wells theorem, the gereralisation being due to Robert Wells (*Cobordism Groups of Immersions*, Topology 5 (1966)). The same is also true for connected Thom spaces $T\xi$ other than $S^r$, giving bordism of immersions with other structures on their normal bundles, eg orientations.


The idea of the correspondence is as follows. Given a map $$(M\times \mathbb{R}^n)\_ + \simeq \Sigma^n M\_{+} \to \Sigma^n T\xi = T(\xi\oplus \varepsilon^n)$$ we can make it transverse to the zero section in $\xi\oplus\varepsilon^n$, and therefore obtain a codimension $r+n$ embedding in $M\times\mathbb{R}^n$ with $n$ independent normal vector fields. By performing an isotopy we can make these normal vector fields point parallel to the $\mathbb{R}^n$ directions, and thus obtain an embedding which projects down to a codimension $r$ immersion in $M$ (this is the content of the *Compression Theorem* of Rourke and Sanderson). To go the other way, start with an immersion $N\looparrowright M$ whose normal bundle is classified by a map $\nu\to\xi$ and compose with the embedding $M\times\{0\}\hookrightarrow M\times\mathbb{R}^n$. If $n$ is sufficiently large then this will be regularly homotopic to an embedding whose normal bundle is classified by $\nu\oplus\varepsilon^n\to \xi\oplus\varepsilon^n$. Now take the collapse map of this.


