Hello everyone,
in [1] I found some explicit generators of homotopy groups of unitary groups, for example
π3[SU2]:
[z1z2]→[z1−¯z2z2z1]
and
π5[SU3]:
[z1z2z3]→[z1+ˉz3z2−ˉz23−ˉz2+ˉz3ˉz1z22z1−z2ˉz3z3+z2ˉz1−z3+ˉz1z2−ˉz2−ˉz1ˉz3ˉz21].
I am in search of explicit 2x2 and 3x3 unitary matrices of this form for π4[SU2] and π5[SU2], as well as πn[SU2/3] (n>5), but haven't been able to find them (maybe they don't exist). If anyone knows of a paper containing them or an easy method of constructing them, I would greatly appreciate it.
Many thanks,
Pascal
[1]: http://arxiv.org/abs/math/0301192v1