It is a known result that based point maps $f:S^2\to S^2$ are classified by their degree. That is, by the induced map at $\pi_2$-level $f_{*2}:\pi_2(S^2)\to\pi_2(S^2)$ (the subindex $*2$ just mean it is the induced map by $f$ at $\pi_2$-level). However, I am interested in knowing which is the relationship (if there is any) between the induced maps $f_{*2}:\pi_2(S^2)\to\pi_2(S^2)$ and $f_{*3}:\pi_3(S^2)\to\pi_3(S^2)$. For instance, if the former is the zero map it means that $f$ is null-homotopic and therefore the latter should also be the zero map. But, it they are not zero, how are they related? Thanks in advance.

It is a known result that based point maps $f:S^2\to S^2$ are classified by their degree. That is, by the induced map at $\pi_2$-level $f_{*2}:\pi_2(S^2)\to\pi_2(S^2)$ (the subindex $*2$ just means it is the induced map by $f$ at $\pi_2$-level). However, I am interested in knowing which is the relationship (if there is any) between the induced maps $f_{*2}:\pi_2(S^2)\to\pi_2(S^2)$ and $f_{*3}:\pi_3(S^2)\to\pi_3(S^2)$. For instance, if the former is the zero map it means that $f$ is null-homotopic and therefore the latter should also be the zero map. But, it they are not zero, how are they related? Thanks in advance.

It is a known result that based point maps $f:S^2\to S^2$ are classified by their degree. That is, by the induced map at $\pi_2-$level $f_{*2}:\pi_2(S^2)\to\pi_2(S^2)$ (the subindex $*2$ just mean it is the induced map by $f$ at $\pi_2$-level). However, i am interested in knowing which is the relationship (if there is any) between the induced maps $f_{*2}:\pi_2(S^2)\to\pi_2(S^2)$ and $f_{*3}:\pi_3(S^2)\to\pi_3(S^2)$. For instance, if the former is the zero map it means that $f$ is nullhomotopic and therefore the latter should also be the zero map. But, it they are not zero, how are they related? Thanks in advance.

It is a known result that based point maps $f:S^2\to S^2$ are classified by their degree. That is, by the induced map at $\pi_2$-level $f_{*2}:\pi_2(S^2)\to\pi_2(S^2)$ (the subindex $*2$ just mean it is the induced map by $f$ at $\pi_2$-level). However, I am interested in knowing which is the relationship (if there is any) between the induced maps $f_{*2}:\pi_2(S^2)\to\pi_2(S^2)$ and $f_{*3}:\pi_3(S^2)\to\pi_3(S^2)$. For instance, if the former is the zero map it means that $f$ is null-homotopic and therefore the latter should also be the zero map. But, it they are not zero, how are they related? Thanks in advance.

It is a known result that based point maps $f:S^2\to S^2$ are classified by their degree. That is, by the $f_{*2}:\pi_2(S^2)\to\pi_2(S^2)$. However, i am interested in knowing which is the relationship (if there is any) between the induced maps $f_{*2}:\pi_2(S^2)\to\pi_2(S^2)$ and $f_{*3}:\pi_3(S^2)\to\pi_3(S^2)$. For instance, if the former is the zero map it means that $f$ is nullhomotopic and therefore the latter should also be the zero map. But, it they are not zero, which is their relationship? Thanks in advance.

It is a known result that based point maps $f:S^2\to S^2$ are classified by their degree. That is, by the induced map at $\pi_2-$level $f_{*2}:\pi_2(S^2)\to\pi_2(S^2)$ (the subindex $*2$ just mean it is the induced map by $f$ at $\pi_2$-level). However, i am interested in knowing which is the relationship (if there is any) between the induced maps $f_{*2}:\pi_2(S^2)\to\pi_2(S^2)$ and $f_{*3}:\pi_3(S^2)\to\pi_3(S^2)$. For instance, if the former is the zero map it means that $f$ is nullhomotopic and therefore the latter should also be the zero map. But, it they are not zero, how are they related? Thanks in advance.