Counting the ways a sphere maps to a space

Besides homology, the other major measuring tool in algebraic topology is that of the homotopy groups \({\pi_{n}\left(X\right)}\). These functors count the number of non-homotopic maps existing from \({S^{n}}\) to the space \({X}\). Recalling the definition of homotopic, we see that this is essentially a count of the classes of mapped spheres that cannot be deformed into each other.

Although the construction of the homotopy groups is much simpler than that of the homology groups, what they actually measure in the space is less easily described in an intuitive way, for example in terms of “holes.” This is apparent from the fact that like the homology groups, \({\pi_{i}\left(S^{n}\right)=\mathbb{Z}}\) for \({i=n}\) and vanishes for \({i<n}\); however for \({i>n}\) we have \({H_{i}(S^{n})=0}\) as we would intuitively expect, while \({\pi_{i}\left(S^{n}\right)}\) yields a complicated table of groups that comprises an area of active research. This is representative of the fact that the homotopy groups are in general much harder to compute than the homology groups.

An Illustrated Handbook