In this lecture we finally define the higher homotopy groups $\pi_n(X)$ all $n \ge 0$. We prove that they are always abelian if $n \ge 2$.

We then prove that the homotopy groups form a local system of groups. This means the following: let $\Pi(X)$ denote the fundamental groupoid of $X$ (cf. Lecture 3). Then there is a functor $T \colon \Pi(X) \to \mathsf{Groups}$ which on objects of $\Pi(X)$ (that is, points in $X$) is given by $x \mapsto \pi_n(X,x)$.

We use this to deduce that if $X$ is path connected then the groups $\pi_n(X,x)$ do not depend on $x$, and that if $X$ and $Y$ have the same homotopy type then they have the same homotopy groups.

Finally, we take a brief look at the nightmare that is $\pi_n(S^k)$ for general $n$ and $k$!