# The map $S_n \to \Pi_n / J_n$

Let $S_n$ denote the set of all oriented diffeomorphism classes of closed smooth homotopy n-spheres. Let $S_n^{bp}\subseteq S_n$ denote the subgroup represented by homotopy spheres that bound parallelizable maifolds. Let $\Pi_n = \pi_{n+q}(\mathbb S^q)$ denote the stable homotopy group of spheres.

In Differential Topology 46 years later Milnor states that there is an exact sequence $$0 \to S_n^{bp} \to S_n \to \Pi_n / J_n$$ where $J_n$ is the Whitehead homomorphism.

What is the map $S_n \to \Pi_n / J_n$ and where can I read about it?

The map $S_n \longrightarrow \Pi_n/\mathrm{Im}(J_n)$ is defined as follows: Let $\Sigma \in S_n$ be a homotopy $n$-sphere. By the Whitney embedding theorem, we may embed $\Sigma$ in $S^{n+k}$ for $k$ large enough. Choosing a framing of $\Sigma$ in $S^{n+k}$, we may apply the Pontryagin-Thom construction to $(\Sigma, \varphi)$ to get an element of $\Pi_n$. Let $p(\Sigma) \subset \Pi_n$ be the set of all such elements obtained by considering all possible framings of $\Sigma$ in $S^{n+k}$. One can show that for any homotopy $n$-sphere, $p(\Sigma)$ is a coset of the subgroup $p(S^n)$ in $\Pi_n$. But $p(S^n)$ is really just $\mathrm{Im}(J_n)$. Hence the assignment $$\Sigma \mapsto p(\Sigma)$$ defines a homomorphism $S_n \longrightarrow \Pi_n/\mathrm{Im}(J_n)$.