Draft:Homotopy theory

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

Homotopy groups of spheres

Main article: Homotopy groups of spheres

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

$*={\pi }_1ℝ\to {\pi }_1{S}^1\to {\pi }_{0}F\to {\pi }_0{S}^1=*.$

Thus, ${\pi }_1{S}^1\simeq {\pi }_{0}F=\mathbb{Z}.$

We have:

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

This can be seen by the smooth or simplicial approximation theorem. Indeed, given a $f:S^k\to S^n$, varying it through homotopy, we can assume

Shape theory

Main article: Shape theory

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.