# 4. The fundamental group

In this lecture we defined the fundamental group. It is perhaps now appropriate to say a few words about the history of algebraic topology, since it basically started with the discovery of the fundamental group.

##### Some history:

Algebraic Topology began life as a subject with a paper by Henri Poincaré in 1895 called *Analysis Situs*. In this paper he single-handedly invented both the fundamental group and homology, and introduced a new way of thinking: that topological problems could be solved via algebraic invariants.

Poincaré did *not* define $ \pi_1$ as a functor from $\mathsf{Top}$ to $\mathsf{Groups}$ ! In fact, it took another fifty years after Poincaré until category theory was even invented!

Whilst *Analysis Situs* was a wonderful piece of work, it was sadly full of errors. Ultimately Poincaré wrote five “supplements” to *Analysis Situs* that successively rectified some of the problems.

Poincaré recognised that his work left many open problems, including for instance the invariance of domain problem—that $ \mathbb{R}^n$ is not homeomorphic to $ \mathbb{R}^m$ for $n \ne m$. This was proved by Brouwer in 1911. However perhaps the greatest problem Poincaré left open was the one he originally thought trivial.

Later on in the course we will prove that $ \pi_1(S^n) $ is trivial for all $ n \ge 2$. Poincaré knew this too, and to begin with—in Analysis Situs—he thought it was “obvious” that the converse was also true: if $M$ is a closed simply connected manifold with trivial fundamental group then $M$ is homeomorphic to a sphere.

Fast forward to Supplement #2, and Poincaré realised this was false. So he came up with a new statement: if $M$ is a closed manifold with trivial homology then $M$ is homeomorphic to a sphere. Unfortunately this was also false (there exist so-called homology spheres—manifolds that have the same homology as a sphere but are not homeomorphic to spheres).

By the time he got to Supplement #5, he hit upon a better conjecture: if $M$ is a closed three-dimensional manifold with trivial fundamental group then $M$ is homeomorphic to the sphere $S^3$. He remarked that he would not investigate this conjecture since it would “carry him too far away”.

This conjecture remained unsolved for over a hundred years. It became known as the Poincaré Conjecture, and was one of the seven Clay Millennium Mathematical Prizes. It was finally solved by Grigori Perelman in 2003.

*Analysis Situs*, along with its five supplements,* *was translated into English by John Stillwell in 2009. You can read it online here.

*Finally: *Last year I taught a course on Dynamical Systems. Throughout the year I (repeatedly) mentioned that Dynamical Systems was essentially created by Poincaré during his work on the three-body problem in the 1880-1890s. So Poincaré created not just one but *two* new fields of geometry! (In fact, Poincaré was pretty much an expert in *all* areas of mathematics that existed during his lifetime.)