This draft page will be used to work out sections on the homotopy groups of spheres and shape theory

First of all, we have:

${\displaystyle \pi _{k}S^{n}=0}$ for ${\displaystyle k<n}$.

This can be seen by the smooth or simplicial approximation theorem. Indeed, given a ${\displaystyle f:S^{k}\to S^{n}}$, ${\displaystyle f}$ is homotopic to a map that is not surjective (all the maps involved are based). But a sphere with a point removed is contractible.

Let ${\displaystyle p:\mathbb {R} \to S^{1}}$ be the universal cover; explicitly, ${\displaystyle p(t)=e^{2\pi it}}$. Then, taking the long exact sequence of homotopy groups, we get

${\displaystyle *=\pi _{1}\mathbb {R} \to \pi _{1}S^{1}\to \pi _{0}F\to \pi _{0}S^{1}=*}$

where F is a fiber. Thus, ${\displaystyle \pi _{1}S^{1}\simeq \pi _{0}F=\mathbb {Z} }$, since F is a lattice. Using the Hopf bundle ${\displaystyle S^{1}\to S^{3}\to S^{2}}$, by a similar calculation, we also get:

${\displaystyle \pi _{2}S^{2}=\mathbb {Z} .}$

Alternatively, we can also use the Hurewicz theorem to do calculations. Indeed, according to the theorem, ${\displaystyle \pi _{k}S^{n}\simeq \operatorname {H} _{k}(S^{n};\mathbb {Z} )}$ and the latter

There is a homomorphism called the J-homomorphism

${\displaystyle \pi _{k}SO(n)\to \pi _{n+k}S^{n}.}$

Homotopy theory works the best with spaces with nice local behaviors; e.g., CW complexes or absolute neighborhood retracts. Shape theory extends homotopy theory to spaces with poor local behaviors. A canonical example is a Warsaw circle.