Homotopy $\pi_4(SU(2))=\mathbb{Z}_2$ Recently I read a paper using $$\pi_4(SU(2))=\mathbb{Z}_2.$$ Do you have any visualization or explanation of this result? 
More generally, how do physicists understand or calculate high dimension homotopy group?
 A: It is very hard to visualize these homotopy classes, since they correspond to maps $S^4\rightarrow SU(2)\approx S^3$. The homotopy groups of spheres (and any other space) are typically very difficult to calculate in generality and physicists typically ask mathematicians. But there exist simple results in the so-called "stable range" where there is a regular structure (Bott periodicity, $\pi_k(U(N)) = \pi_{k+2}(U(N))$ for large enough $N$), and there exist tools to calculate homotopy groups of certain spaces, such as the long exact sequence of a fibration.
For the case of spheres see the table on wikipedia, where the chaotic behavior is clear and $\pi_4(S^3)$ is listed. There is a very good review by Mermin (1) where you can learn how to visualize and calculate simple homotopy groups.
(1): Rev. Mod. Phys. 51, 591–648 (1979)
A: Since $SU(2)$ is topologically a three-sphere $S^3$, you can begin by investigating the homotopy groups of spheres.  Unfortunately, although there are some regular results, such as $\Pi_n(S^n)=\mathbb{Z}$, and $\Pi_m(S^n)=0$ for $m<n$, I don't think there is a single method to calculate $\Pi_m(S^n)$ for $m>n$.  Individual results for $m>n$ are chaotic.  So, I think the answer to your last question is "they would ask a mathematician!", because this (algebraic topology) is a very large topic.
For your specific case, there is a reference given on math overflow, but I don't have the book unfortunately.
A: This question has been posted also at https://mathoverflow.net/questions/115866/homotopy-pi-4su2z-2 with both geometric and algebraic (that was mine!) type of answers. The geometric answers tell of Pontjyagin's method of constructing explicit representations of maps to spheres. The algebraic methods tells of the answer from a general theorem which gives some results on homotopy groups of complexes, and when the conditions under which it works are satisfies, gives detailed and calculable algebraic answers. 
