# Mathematics Prelims

## March 18, 2009

### The Fundamental Group, Part II – The Basepoint

Filed under: Topology — cjohnson @ 2:07 pm

We constructed the fundamental group by looking at the homotopy classes of loops with a fixed basepoint.  The question that naturally arises is how does the fundamental group change if we move the basepoint.  To begin let’s suppose we’re in a path connected space $X$ and we have two basepoints we’re concerned about, say $x_0$ and $x_1$.  Since $X$ is path connected, there exists some path $f$ from $x_0$ to $x_1$.  We can use this to convert $x_1$ loops into $x_0$ loops.

What we’ll do is take a loop, say $g$, based at $x_1$ and use our path to get from $x_0$ to $x_1$, follow the loop, then take the reverse path back to $x_0$.  This is illustrated in the image below (adapted from an image in Hatcher’s book).

This new path $f * g * \overline{f}$ is a loop based at $x_0$.  Now, if two $x_1$-loops, say $g_1$ and $g_2$ are homotopic, then so are the associated $x_0$-loops: $g_1 \simeq g_2 \implies f * g_1 * \overline{f} \simeq f * g_2 * \overline{f}$.

If we now construct a map $\beta_f : \pi_1(X, x_1) \to \pi_1(X, x_0)$ by setting $\beta_f [g] = [f * g * \overline{f}]$ we see that the map will be well-defined since our construction of the $x_0$-loops respects homotopy.  But what happens when we concatenate two $x_1$-loops?

Suppose $g$ and $h$ are loops based at $x_1$.  We want to see what this map does to the homotopy class of $g * h$.

$\displaystyle \beta_f([g] \cdot [h]) = \beta_f[g * h]$

$\displaystyle = [f * g * h * \overline{f} ]$

$\displaystyle = [f * g * \overline{f} * f * h * \overline{f} ]$

$\displaystyle = [f * g * \overline{f}] \cdot [f * h * \overline{f}]$

$\displaystyle = \beta_f [g] \cdot \beta_f [h]$

So our $\beta_f$ map is in fact a homomorphism.  Notice that $f * g * \overline{f} \simeq f * h * \overline{f}$ implies $g \simeq h$, which means that our map is injective.

To show the map is surjective, let $h$ be an $x_0$-loop, then $\overline{f} * h * f$ is an $x_1$-loop.  Applying $\beta_f$ we have $\beta_f[\overline{f} * h * f] = [f * \overline{f} * h * f * \overline{f}] = [h]$, and so our map is surjective.  Together with injectivity above we have shown that in a path connected space the fundamental group of $\pi_1(X, x_0)$ and $\pi_1(X, x_1)$ are isomorphic.  We then simply write $\pi_1(X)$ since the choice of basepoint is basically irrelevant.

If our space is not path connected, but $x_0$ and $x_1$ are in the same path component, the above gives us that $\pi_1(X, x_0) \cong \pi_1(X, x_1)$.