Surjectivity of J homomorphism Can somebody say something about surjectivity of the $J$ homomorphism $J_{7,7}$ : $\pi_{7}$SO($7$) $\rightarrow$ $\pi_{14}(S^7)$ ? Husemoller in Fibre Bundles says this :
Where $M_{n}^{1}$ denotes component of $\text{Map}_{0}(S^{n-1},S^{n-1})$ consisting of maps of degree $1$. I do not know how to proceed after this. Also is there any information about unstable $J$ homomorphisms ?

It is not surjective.  More specifically, using a table of homotopy groups of spheres on Wikipedia, I can prove that this map is not surjective.  But I cannot speak to the veracity of the table.
The key observation is Proposition 6.4 which tells you that $J = E\circ \theta \circ\epsilon_\ast$.  Note, then, that if $E$ is not surjective, $J$ cannot possible be surjective.
According to this Table of homotopys groups of spheres, $\pi_{13}(S^6)\cong \mathbb{Z}_{60}$ while $\pi_{14}(S^7) \cong \mathbb{Z}_{120}$.  Thus, $E$ is obviously not surjective: its image consists of at most half the elements in $\pi_{14}(S^7)$.
