Today we proved that the fundamental group is not a complete waste of time, by checking that at least one topological space has a non trivial $\pi_1$. Since (on Problem Sheet B) we know that $\pi_1(X,p) \cong [(S^1,1),(X,p)]$ (the morphism space in $\mathsf{hTop}_*$), it seems reasonable that a good space to try and compute $\pi_1$ is for $S^1$ itself. The conclusion:

$$\pi_1(S^1) \cong \mathbb{Z},$$

is probably not that surprising to any of you (I mean, if you think about it, what else could it possibly be? 🤔)

We concluded the lecture by briefly mentioning universal properties, which, roughly speaking, are a fancy way of requiring a diagram to "commute as efficiently as possible." We will see many examples of this as time goes by. We then defined the notion of a pushout in a category. A pushout is a special case of the more general notion of a colimit. (Namely, a pushout is a colimit of the diagram $\bullet \leftarrow \bullet \rightarrow \bullet$.) We will study colimits later in the course.

##### Remark

If you are familiar with covering spaces, you will no doubt realise that the main result (Proposition 5.2) established the homotopy lifting property for covering spaces in the special case of the cover  $\mathbb{R} \to S^1$.