I am a physics student, recently I read a paper using Homotopy $\pi\_4(SU(2))=Z\_2$, I guess mathematicians have some **visualization or explanation** of this result. So I come here ask for help. 


CROSS-POST from <https://physics.stackexchange.com/questions/46284/homotopy-pi-4su2-z-2>



Homotopy $\pi_4(SU(2))=Z_2$
This calculation of $\pi\_4(S^3)$ is also obtained in the paper R. Brown and J.-L. Loday, Topology, 26 (1987) 311-334, and also available [here](http://groupoids.org.uk/pdffiles/RB-Loday1.pdf). In that paper, $S^3$ is regarded as the double suspension $SS$ of the circle $S^1$, which is itself seen as an Eilenberg-Mac Lane space $K(\mathbb Z,1)$. We obtain in Proposition 4.10 a determination of $\pi\_4$ of the double suspension $SS$ of a $K(G,1)$ for any group $G$ as the kernel of a morphism $G \tilde{\wedge} G \to G$ defined by the commutator map, where the group $G \tilde{\wedge} G$ is the quotient of the free group on the set $G \times G$ by a set of relations satisfied by commutators. Hence the result for $\pi\_4 SK(G,1)$ is easy to calculate if $G$ is abelian: in fact in that case it is $G \otimes G$, the tensor product of abelian greoups, factored by the relations $g \otimes h + h \otimes g$ for all $g,h \in G$. 


Part of the intuition behind this is that the suspension $SX$ of a space $X$ is regarded as the union $C^+X \cup C^- X$ of two cones with intersection $X$, and this union is one to which our van Kampen type theorem for squares of spaces can apply. In fact we are dealing with the triad $(SX; C^+X, C^-X) $, which is the union of two triads $(C^+X;C^+X,X)$, $(C^- X; C^-X,X)$, and the union of these two "trivial" triads creates something in dimension $3$. If $\pi\_2 X=0$, this gives a complete determination of $\pi\_3 SX$, and further work gives a result on the double suspension in the given case. 


So the intuition is that in homotopy theory, identifications in low dimensions have high dimensional homotopy implications, and to cope with this for gluing purposes we need algebraic structures with structures in a range of dimensions. The hope is that someday such structures will have applications in physics! 


