Homotopy Groups of Spheres - General Theory

General Theory

As noted already, when i is less than n, π_i(Sn) = 0, the trivial group (Hatcher 2002). The reason is that a continuous mapping from an i-sphere to an n-sphere with i < n can always be deformed so that it is not surjective. Consequently, its image is contained in Sn with a point removed; this is a contractible space, and any mapping to such a space can be deformed into a one-point mapping.

The case i = n has also been noted already, and is an easy consequence of the Hurewicz theorem: this theorem links homotopy groups with homology groups, which are generally easier to calculate; in particular, it shows that for a simply-connected space X, the first nonzero homotopy group π_k(X), with k > 0, is isomorphic to the first nonzero homology group H_k(X). For the n-sphere, this immediately implies that for n > 0, π_n(Sn) = H_n(Sn) = Z.

The homology groups H_i(Sn), with i > n, are all trivial. It therefore came as a great surprise historically that the corresponding homotopy groups are not trivial in general. This is the case that is of real importance: the higher homotopy groups π_i(Sn), for i > n, are surprisingly complex and difficult to compute, and the effort to compute them has generated a significant amount of new mathematics.