Since a one-simplex is homeomorphic to an interval, it will no doubt surprise none of you that there is an intimate relation between $\pi_1$ and $H_1$. Of course they cannot be the same, since the latter is abelian (by definition) and the former may not be (cf. the Klein bottle).

Luckily there is a nice algebraic trick to convert a group $G$ into an abelian group $G^{ \mathrm{ab}}$, by quotienting out all the pesky non-abelian elements. This process $G \mapsto G^{ \mathrm{ab}}$ is called the abelianisation of $G$.

Today we proved that by applying this "abelian-killer" process to $\pi_1$, we obtain $H_1$. This result is called the Hurewicz Theorem, even though it was actually proved by Poincaré[1].

  1. To be fair, Hurewicz did prove a massive generalisation of this result, which relates the higher homotopy groups to the higher homology groups. We'll cover this in Algebraic Topology II next semester. ↩︎

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