representation for elements in $\pi_3(S^3)$

Question

I want to know if the map between $\mathbb{Z}$ and $\pi_3(S^3) $ given by $a \mapsto (x \mapsto x^a)$, where $S^3$ is seen as a subset of $\mathbb{H}$ with its multiplication, gives an isomorphism just like the case $\pi_1(S^1) $?

Answer 1

The answer is yes: Since quaternionic multiplication equips $S^3$ with an H-space structure, the sets $[X, S^3]_*$ inherit a multiplication (which we denote by $\mathbin{\cdot_{\mathbb{H}}}$) for all $X$, and specializing to $X = S^n$ this agrees with the usual multiplication on homotopy groups. Now $[\mathrm{id}_{S^3}]$ is a generator for $\pi_3(S^3)$ and $\mathrm{id}_{S^3} \mathbin{\cdot_{\mathbb{H}}} \mathrm{id}_{S^3} = (x \mapsto x \cdot x = x^2)$, whence $\mathrm{id}_{S^3}^n = (x \mapsto x^n)$ represents $n \cdot [\mathrm{id}_{S^3}]$ in $\pi_3(S^3)$ and your map is an isomorphism.