Thanks freikugel. It's definitely a partial answer and very useful. I found it quite funny that I used the Cayley graph in MartianInvader's avatar to prove that the higher homotopy groups of the wedge of two circles (and similarly for n circles) are all 0 due to its universal cover (the Cayley graph of its fundamental group) being contractible
It does seem like covering spaces (and fibrations in general) and the long exact sequence of homotopy groups give a powerful method for finding a lot of higher homotopy groups for spaces. So far I've used the method to show that πn
)=Z for n>1, to show that πn+k
) for n>1, and that πk
)=0 for k>1. I've also used the Hopf fibration to show that πn
) for n>2 and a few other simple results.
I wonder if anyone could suggest some other examples of spaces for which the higher homotopy groups can be worked out this way. Just the spaces of course (and possibly the fibration if it's a non-trivial map like the Hopf fibration) as this work will hopefully lead to an assessed project.