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

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

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

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

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