Definition 1.9.
The group of homotopy classes of loops at \(x_0 \in X\) under \(*\) is denoted \(\pi_1(X, x_0)\) and called the fundamental group of \(X\) based at \(x_0\text{.}\)
Section 1.2 The Fundamental Group
A path that starts and ends at the same point is called a loop. Proposition 1.7 shows that homotopy classes of loops based at a fixed point \(x_0\) form a group under \(*\text{.}\)
Definition 1.10.
Let \(\alpha\) be a path in \(X\) from \(x_0\) to \(x_1\text{.}\) The map
\begin{equation*}
\hat{\alpha} \colon \pi_1(X, x_0) \to \pi_1(X, x_1), \quad [f] \mapsto [\bar{\alpha}] * [f] * [\alpha]
\end{equation*}
is called the basepoint-change isomorphism along \(\alpha\text{.}\)
Exercise 1.11.
Show that \(\hat{\alpha}\) is a well-defined group isomorphism.
A consequence of Exercise 1.11 is that if \(X\) is path-connected then all groups \(\pi_1(X, x_0)\) are isomorphic. However, the isomorphism may depend on the choice of path \(\alpha\text{,}\) so there is no canonical identification.
Exercise 1.12.
Show that the isomorphism is independent of choice of path if and only if \(\pi_1(X,x_0)\) is abelian.
Definition 1.13.
A space \(X\) is simply connected if it is path-connected and \(\pi_1(X, x_0) = 0\) for some (hence every) \(x_0\text{.}\)
Proposition 1.14.
A space \(X\) is simply connected if and only if there is a unique homotopy class of paths connecting any two points.
Proof.
If \(X\) is simply connected it is path-connected, so paths between any two points exist. For uniqueness, let \(f\) and \(g\) be paths from \(x_0\) to \(x_1\text{.}\) Then \(f * \bar{g}\) is a loop at \(x_0\text{,}\) hence null-homotopic, so
\begin{equation*}
f \simeq f * \bar{g} * g \simeq c_{x_0} * g \simeq g.
\end{equation*}
Conversely, if there is a unique homotopy class of paths between any two points, then setting \(x_0 = x_1\) shows every loop is null-homotopic.
The fundamental group is a topological invariant in a functorial way. Note that it takes values in groups (not necessarily abelian), giving a functor
\begin{equation*}
\pi_1 \colon \mathbf{Top}_* \to \mathbf{Grp}
\end{equation*}
from the category of based topological spaces to the category of groups.
Proposition 1.15.
Let \(\phi \colon (X, x_0) \to (Y, y_0)\) be a based continuous map. The map
\begin{equation*}
\phi_* \colon \pi_1(X, x_0) \to \pi_1(Y, y_0), \quad [f] \mapsto [\phi \circ f]
\end{equation*}
is a well-defined group homomorphism, called the induced homomorphism, satisfying:
-
\((\mathrm{Id}_X)_* = \mathrm{Id}_{\pi_1(X,x_0)}\text{,}\)
-
\((\psi \circ \phi)_* = \psi_* \circ \phi_*\) for composable based maps.
Exercise 1.16.
Prove Proposition 1.15.
