The J-homomorphism is a well-known and classical map $\pi_n (O(k)) \to \pi_{n+k} (S^k)$, or after stabilizing with respect to $k$, a map $J_n:\pi_n (O) \to \pi_{n}^{st}$, from the stable homotopy of orthogonal groups to stable homomotopy of spheres. The main results on $J_n$ were proven by Adams in a classic series of four papers.

The results are: the image of $J_{4r-1}$ (in which case the source group is $\mathbb{Z}$) is cyclic of order $den (\frac{B_r}{4r})$ (denominator of Bernoulli numbers). For $n\equiv 0,1 \pmod 8$, $J_n$ is injective (the source is $\mathbb{Z}/2$). Another theorem by Adams is that the unit map $\pi_{n}^{st} \to \pi_n (BO)= \pi_{n-1} (O)$ hits all $\mathbb{Z}/2$-groups in the image.

These results play an important role in differential topology (for example in the classification of exotic spheres), which is why from time to time, I struggle to understand these results. But I am rather foreign to stable homotopy theory and I am scared away by this battle with homological algebra and stable homotopy theory and never manage to get the main points from Adams' papers.

However, for the case $n=4r-1$, there is a version of the proof without leaving the mathematical terrain I am used to navigate in. There are two parts: proving that the image of $J$ has \it{at least} the order $den (\frac{B_r}{4r})$ requires an invariant an a device for its computation. The invariant is the $e$-invariant $e: \pi_{4n-1}^{st} \to \mathbb{Q}/\mathbb{Z}$ and the computation of $e \circ J$ is done using characteristic classes. All this is well-explained in Hatcher's book project on $K$-theory, with a hands-on definition of the $e$-invariant.

Proving that the image of $J$ has \it{at most} the order $den (\frac{B_r}{4r})$ requires a construction of a nullhomotopy. It follows from the Adams conjecture and some Bernoulli numerology. Besides the first proofs of the Adams conjecture by Quillen and Sullivan, there exist two proofs which I understand (by Becker-Gottlieb and a simplification of it by E. Brown, which I wrote up some years ago).

Here are my questions:

1. Is there an argument for the injectivity of $J_{n}$, $n \equiv 0,1 \pmod 8$, which is similarly direct as the argument in Hatcher's book?

2. The $J$-homomorphism gives a map of spectra $\Sigma^{-1} KO \to S$. What is the composition with the unit map $S \to KO$, and what is a low level explanation for it?

3. Is there a low-level description for the image of the unit map $S \to KO$ in homotopy?

4. Or do have to learn it the hard way?