Calculating the higher homotopy groups

Calculating the homotopy groups beyond the fundamental group is usually much more difficult, since Van Kampen’s theorem doesn’t hold and excision type theorems are much weaker. An important tool is the relationship between homotopy and homology groups. For example, the Hurewicz theorem states that for a simply connected space, the first nonzero homotopy group \({\pi_{n}}\) is isomorphic to the first nonzero homology group \({H_{n}}\) in the same dimension.

Some specific results are that for \({n>1}\), \({\pi_{n}(\mathbb{R}\textrm{P}^{d})=\pi_{n}(S^{d})}\), while the higher homotopy groups vanish for the Klein bottle and the torus of any dimension \({T^{n}}\).

An Illustrated Handbook