I'm unsure whether this question is appropriate for mathoverflow, so feel free to criticize.

All manifolds are closed, smooth and have dimensions $n\ge 5$.

The Atiyah-Shapiro-Bott-Orientation gives a ring homomorphism $$\alpha\colon\Omega_*^{spin}\rightarrow KO^{-*}(pt),$$ from the spin-bordism ring to real K-theory, whose vanishing is a necessary condition for a spin manifold admitting a metric of positive scalar curvature. Recall $$KO^{-n}(pt)\cong \begin{cases} \mathbb{Z} &\mbox{if } n \equiv 0,4 (8)\\ \mathbb{Z}/2\mathbb{Z} & \mbox{if } n \equiv 1,2(8) \\ 0 &\mbox{if } n \equiv 3,5,6,7 (8). \end{cases}.$$ In dimensions $n\equiv0,4(8)$, $\alpha$ is just the $\hat{A}$-genus (respectively twice of it).

Considering the spin-cobordism class of homotopy spheres, the $\alpha$-invariant induces homomorphisms $$\beta_n\colon\Theta_n\rightarrow KO^{-n}(pt),$$ ($\Theta_n$ is the group of $n$-dimensional homotopy spheres), which are zero in dimensions $n\equiv 3,5,6,7 (8)$ for trivial reasons.

In the other dimensions, we have

- $\beta_n$ is zero in dimensions $n\equiv0,4(8)$, what is equivalent to the vanishing of the $\hat{A}$-genus for all homotopy spheres.
- $\beta_n$ is surjective in dimensions $n\equiv1,2(8)$.

I am trying to better understand these two claims. Lawson and Michelson simply write in their book "Spin Geometry", that (2) follows from "deep work of Adams and Milnor".

**Since I am absolutely not an expert in this field, can someone elaborate a bit (more than "It follows from the work of Adams and Milnor.", what is not really helpful for me.) on what I really need to prove (2) and how one can place it in a wider context?**

**How can one prove (1) and is there a reference for it?**

I searched the literature without good results, so simply giving references might also answer my question.