TITLE: Homotopy groups other than $\pi_1$ : what are they good for?
QUESTION [7 upvotes]: Homotopy groups $\pi_k$ were introduced before the homology groups $H_k$ in the 1st topology book I read and the 1st topology course I took. Later on $\pi_1$, $H_k$, and $H^k$ appeared in numerous contexts in a variety of subjects. 
On the other hand, I never encountered $\pi_k$, $k\ge 2$ beyond the topology textbooks, with the exception of the $\pi_3(S^2)$ curiosity, which IMHO is just an algebraic sideshow for Hopf fibration.
Therefore the question: are homotopy groups $\pi_k$, $k\ge 2$ useful for something that cannot done with other invariants such as $H_k$?

REPLY [7 votes]: Higher homotopy groups arise in the study of topological defects in physics.
Many physical systems are described by an "order parameter", a manifold valued field on a region of space, $f:U\subset \mathbb{R}^3\rightarrow M$. For example, in some materials there is a continuous (if you don't look too closely) vector valued magnetization property defined at each point in space. The vector has a constant length fixed by the nature of the material. So we have a field $f:\mathbb{R}^3\rightarrow S^2$. In a certain type of liquid crystal the molecules are shaped roughly like line segments so at each point the orientation of the molecules is given by a line. So we have a field $f:\mathbb{R}^3\rightarrow \mathbb{R}P^2$. Dislocations in crystals give rise to maps $f:\mathbb{R}^3\rightarrow T^3$ where $T^3$ is the 3-torus.
Sometimes these order parameters have singularities, points or curves where $f$ is no longer defined. These can be studied by looking at the region around the "defect". For example, if $f$ has a point singularity at the origin then we can look at $f$ on a sphere, $S^2$, around the defect. This gives rise to the restriction $f':S^2\rightarrow M$ which gives an element of $\pi_2(M)$. If $f$ is continuous everywhere except at the origin, then continuously moving the sphere outwards from the origin won't change which homotopy class $f'$ represents. So if $f'$ represents a non-trivial homotopy class then in some sense the defect can't be hidden or deformed away and can be detected on the surface of a sphere a long way from the defect itself. These phenomena have a variety of physical consequences.
(Defects along a line give rise to elements of $\pi_1(M)$ and elements of $\pi_3(M)$ can give rise to a non-localisable "texture defect".)
There is a substantial amount of literature on these defects and they apply to everything from solid and liquid crystals to ferromagnets and cosmic strings.
There is a wikipedia article on topological defects. Here is a paper with some pictures of point defects in liquid crystals.
Higher homotopy groups also arise in the (hot right now) field of topological insulators.