Using van-kampen theorem, Fundamental group of wedge of n-circles is free group on n-generator. But I don't know how to calculate higher homotopy groups of wedge of spaces, in particular circles. I just came to know that there is some result by Milnor regarding higher homotopy groups of sphere,but I doubt myself that I have understood it.
Any help in this regards will be appreciated.
Thanks.
[Math] Higher homotopy groups of wedge of circles.
algebraic-topologyhomotopy-theory
Best Answer
If $X$ is a covering of $Y$, then the higher homotopy groups of $X$ and $Y$ agree. If you like, this is a consequence of the lifting lemmas studied in covering space theory.
One consequence of this is that if the universal cover of a space is contractible, then all of its higher homotopy groups vanish. This is in particular true of a wedge of circles.