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I have a feeling that this is well-known:

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 $M$. Does this respect suspension? If so, why? I am willing to settle for such a statement on $M\approx S^{n+r}$, i.e. where we suspend $(n+r)$-homotopy classes of the $r$-sphere to $(n+r+1)$-homotopy classes of the $(r+1)$-sphere.
So in particular I would like to know what 'suspension' means on the framed manifold/cobordism side.

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The suspension of a manifold is a (topological) manifold only if $M$ is a sphere. So it is not clear what a submanifold of a suspension is. – user17786 Nov 21 '12 at 0:57
up vote 2 down vote accepted

Let $f: S^k \longrightarrow S^r$ be a smooth map. Then its suspension $$Sf: S^{k+1} \longrightarrow S^{r+1}$$ is smooth away from the basepoints (i.e. the north of south poles of $S^{k+1}$ and $S^{r+1}$). Let $x \in S^r$ be a regular value of $f$ different than the basepoint of $S^r$, and let $A = f^{-1}(x)$ be the framed submanifold of $S^k$ associated to $f$. We clearly have that $$A = (Sf)^{-1}(x) \subset S^k \subset S^{k+1},$$ where we consider $S^k$ as the equator of $S^{k+1}$ and view $x \in S^r$ as lying in the equator of $S^{r+1}$. Now \begin{align*} \nu(A \hookrightarrow S^{k+1}) & = \nu(A \hookrightarrow S^k) \oplus \nu(S^k \hookrightarrow S^{k+1}) \\ & = \nu(A \hookrightarrow S^k) \oplus \varepsilon_A^1, \end{align*} where $\varepsilon_A^1$ is the trivial line bundle over $A$, and similarly \begin{align*} \nu(\{x\} \hookrightarrow S^{r+1}) & = \nu(\{x\} \hookrightarrow S^r) \oplus \nu(S^r \hookrightarrow S^{r+1}) \\ & = \nu(\{x\} \hookrightarrow S^r) \oplus \varepsilon_{\{x\}}^1. \end{align*} Since near the equator we have $Sf = f \times \mathrm{Id}$, $Sf$ preserves the canonical framing of $S^r$ in $S^{r+1}$, and therefore we see that we get a well-defined framing of $A$ in $S^{k+1}$.

From the above, we see that when $M \cong S^k$, the Pontrjagin-Thom construction tells us that suspensions of homotopy classes of maps $f: S^k \longrightarrow S^r$ correspond to framed submanifolds of $S^k$ embedded in the equator of $S^{k+1}$, with its framing Whitney summed with the canonical framing of $S^k$ in $S^{k+1}$.

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This is great! Do you happen to know if this also works for general compact simply-connected oriented 4-manifolds $M$? – Chris Gerig Nov 20 '12 at 18:06

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