Higher Homotopy Groups & Hurewicz

Throw spheres instead of loops: the same construction one dimension up, abelian and almost uncomputable.

The idea

The higher homotopy groups $\pi_{n}(X, x_{0})$ extend the fundamental group to every dimension: where $\pi_{1}$ was built from maps of the circle into $X$, the $n$-th group is built from maps of the $n$-sphere. Nothing in the construction of $\pi_{1}$ used the circle's one-dimensionality.

Definition (Homotopy groups).

Fix basepoints $x_{0} \in X$ and $s_{0} \in S^{n}$. The $n$-th homotopy group $\pi_{n}(X, x_{0})$ is the set of homotopy classes of maps $f \colon S^{n} \to X$ with $f(s_{0}) = x_{0}$, where the homotopies too must hold the basepoint at $x_{0}$ at every instant.

The addition uses a second model. Collapsing the boundary of the cube $I^{n}$ to a point gives an $n$-sphere, so a class may equally be represented by a map of $I^{n}$ sending all of $\partial I^{n}$ to $x_{0}$. In that model, the sum of two classes cuts the cube in half along its first coordinate and runs one map on each half. Only that one coordinate is involved, so every argument that made $\pi_{1}$ a group carries over unchanged, and for $n = 1$ the sum is ordinary concatenation of loops.

Two features are new. For $n \ge 2$ the group is abelian, which is why the operation is written additively. And unlike homology, these groups admit no general algorithm: most of them, even for spheres, remain unknown.

Ways to work on it

• Walkthrough. Build _n(X) from maps of spheres and establish its first computations and properties.
• Proof. Why _n is abelian for n ≥ 2 — shrink the two summands onto small subcubes and slide them past each other, which one dimension cannot do.
• Practice. Compute homotopy groups of spheres and product spaces, and apply the Hurewicz theorem.
• Hardest. Apply the Hurewicz theorem and probe the limits of what homology can detect.

Not sure where to start? Take the ten-question placement test.