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Fixed title (besides being mathjax only, it didn't reflect the question itself); fixed \mathbb{Z} through the question
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edit approved Mar 13, 2016 at 4:52
Silvia Ghinassi
edit approved Mar 13, 2016 at 4:52
Silvia Ghinassi
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Is there a homomorphism between $\pi_8(S^5)=\pi_8$ and $\pi_8(SO(6))=\mathbb{Z}/24$$?

I am currently thinking about a physics model related to framed bordism $\Omega_3^{fr}=Z/24=\pi^s_3$$\Omega_3^{fr}=\mathbb{Z}/24=\pi^s_3$, and the first stable example is $\pi_8(S^5)$, so I was curious about the generator, and happened to see $\pi_8(SO(6))=Z/24$$\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 groupgroups $\pi_n(S^m)$ and $\pi_n(SO(m+1))$ in general?

$\pi_8(S^5)=\pi_8(SO(6))=\mathbb{Z}/24$?

I am currently thinking about a physics model related to framed bordism $\Omega_3^{fr}=Z/24=\pi^s_3$, and the first stable example is $\pi_8(S^5)$, so I was curious about the generator, and happened to see $\pi_8(SO(6))=Z/24$ 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 group $\pi_n(S^m)$ and $\pi_n(SO(m+1))$ in general?

Is there a homomorphism between $\pi_8(S^5)$ and $\pi_8(SO(6))$?

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?

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Yingfei Gu
Yingfei Gu
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$\pi_8(S^5)=\pi_8(SO(6))=\mathbb{Z}/24$?

I am currently thinking about a physics model related to framed bordism $\Omega_3^{fr}=Z/24=\pi^s_3$, and the first stable example is $\pi_8(S^5)$, so I was curious about the generator, and happened to see $\pi_8(SO(6))=Z/24$ 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 group $\pi_n(S^m)$ and $\pi_n(SO(m+1))$ in general?