# 5. The fundamental group of the circle and pushouts

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? ðŸ¤”)

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 Â $\mathbb{R} \to S^1$.

Comments and questions?