I am currently thinking about a physics model related to framed bordism $\Omega\_3^{fr}=\mathbb{Z}/24=\pi^s\_3$, and the first stable example is $\pi\_8(S^5)$, so I was curious about the generator, and happened to see $\pi\_8(SO(6))=\mathbb{Z}/24$ as well.


So my question is: is there a homomorphism sending $\pi\_8(SO(6))$ to $\pi\_8(S^5)$? More generally, what's the relation between homotopy groups $\pi\_n(S^m)$ and $\pi\_n(SO(m+1))$ in general?



Is there a homomorphism between $\pi_8(S^5)$ and $\pi_8(SO(6))$?
There is a fibration $p : SO(m+1) \to S^m$ with fibre $SO(m)$ which induces a long exact sequence in homotopy


$$\dots \to \pi\_n(SO(m)) \to \pi\_n(SO(m+1)) \xrightarrow{p\_\*} \pi\_n(S^m) \to \pi\_{n-1}(SO(m)) \to \dots$$


For your particular question, we have the fibration $p : SO(6) \to S^5$ with fibre $SO(5)$ which gives 


$$\dots \to \pi\_8(SO(5)) \to \pi\_8(SO(6)) \to \pi\_8(S^5) \to \pi\_7(SO(5)) \to \pi\_7(SO(6)) \to \pi\_7(S^5) \to \pi\_6(SO(5)) \to \dots$$


I don't know how to compute these groups myself, but they have been computed by others. From the bottom of page $3$ of [this](http://felix.physics.sunysb.edu/~abanov/Teaching/Spring2009/Notes/abanov-cpA1-upload.pdf) we see that $\pi\_8(SO(5)) = 0$, $\pi\_7(SO(5)) = \mathbb{Z}$, $\pi\_7(SO(6)) = \mathbb{Z}$, and $\pi\_6(SO(5)) = 0$. From [here](https://en.wikipedia.org/wiki/Homotopy_groups_of_spheres) we see that $\pi\_7(S^5) = \mathbb{Z}\_2$ so we have


$$\dots \to 0 \to \pi\_8(SO(6)) \xrightarrow{p\_\*} \pi\_8(S^5) \to \mathbb{Z} \to \mathbb{Z} \to \mathbb{Z}\_2 \to 0 \to \dots$$


As the map $\mathbb{Z} \to \mathbb{Z}\_2$ is surjective, it has kernel $2\mathbb{Z}$. So the map $\mathbb{Z} \to \mathbb{Z}$ must be multiplication by $2$ (or $-2$) which is injective. Therefore the map $\pi\_8(S^5) \to \mathbb{Z}$ is the zero map, so $p\_\*$ is an isomorphism.


The fibration always gives rise to a homomorphism $p\_\* : \pi\_n(SO(m+1)) \to \pi\_n(S^m)$ but it is not necessarily an isomorphism as $p\_\* : \pi\_7(SO(6)) \to \pi\_7(S^5)$ demonstrates.


