# The fundamental group

The simplest homotopy group is the fundamental group $${\pi_{1}\left(X\right)}$$, which counts how many ways a loop can be mapped into a path-connected space $${X}$$. More precisely, we define $${\pi_{1}\left(X,x\right)}$$ to be the set of all homotopy classes of parameterized loop mappings that begin and end at a basepoint $${x}$$. For a path-connected space we can add a path from $${x}$$ to any other point and back as part of the loop, so that $${\pi_{1}\left(X,x\right)}$$ is independent of $${x}$$ and is written $${\pi_{1}\left(X\right)}$$.

The above shows that for a path-connected space $${X}$$, every loop with basepoint $${x}$$ is homotopic to a loop with basepoint $${y}$$, so that $${\pi_{1}\left(X,x\right)=\pi_{1}\left(X,y\right)=\pi_{1}\left(X\right)}$$.

$${\pi_{1}\left(X\right)}$$ becomes a group by defining multiplication as the composition of loop paths from a given basepoint. Technically, this is implemented by dividing the line segment $${I}$$ in half, and applying the first mapping $${f}$$ to the first half segment, $${g}$$ to the second half. Note that this means that homotopies of $${fg}$$ may abandon the midway mapping to the basepoint; for example, the inverse of a path is simply the same path traversed backwards.

Homotopies of loop products $${h=f\cdot g}$$ are only required to keep the endpoints of $${h}$$ mapped to the basepoint, allowing the midpoint $${f_{0.5}=g_{0.5}}$$ to abandon the basepoint. Above we have $${g=f^{-1}}$$, the inverse of $${f}$$, in which case the midpoint leaving the basepoint results in the product $${f\cdot f^{-1}}$$ retracting to a point.

So for example $${\pi_{1}\left(S^{1}\right)=\mathbb{Z}}$$, since a loop can be mapped around the circle any number of times in either direction. For the figure eight $${S^{1}\vee S^{1}}$$, a loop mapped around one circle cannot be homotopically altered to go around the other. Thus each homotopically distinct map can be viewed as a “word” with each “letter” an integer number of loops around each circle, i.e. we arrive at the free product $${\pi_{1}\left(S^{1}\vee S^{1}\right)=\mathbb{Z}*\mathbb{Z}}$$. This example also shows that in general $${\pi_{1}}$$ is non-abelian, unlike $${H_{1}}$$ and as we will see below unlike all other $${\pi_{n}}$$. In fact, $${\pi_{1}}$$ is very general indeed: for any arbitrary group $${G}$$ one can construct a space $${X}$$ for which $${\pi_{1}\left(X\right)=G}$$.

A path-connected space with trivial fundamental group is called simply connected. The name reflects the fact that in such a space there is only one homotopically distinct way to form a path between any two points.

 Δ Note that there are other definitions of “simply connected” in use; some do not require the space to be connected.