This is a slightly revamped version of a question 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 $\Sigma^\infty$?**