Remark 5.8.
Property (1) tells us that \(\beta_\gamma\) is a group homomorphism. Properties (2) and (3) tell us that \(\beta_\gamma\) is an isomorphism, with inverse \(\beta_{\bar\gamma}\text{,}\) where \(\bar\gamma\) is the reverse path.
Section 5.2 Change of Basepoint
If \(X\) is path-connected, \(\pi_n(X, x_0)\) does not depend on the basepoint \(x_0\text{.}\) Let \(\gamma \colon I \to X\) be a path from \(x_0\) to \(x_1\text{.}\) We construct a map
\begin{equation*}
\beta_\gamma \colon \pi_n(X, x_1) \to \pi_n(X, x_0), \qquad [f] \mapsto [\gamma \cdot f],
\end{equation*}
as follows: shrink the domain of \(f\) to a smaller cube inside \(I^n\) and insert the path \(\gamma\) radially on the boundary of the cube. The resulting map \(\gamma \cdot f\) agrees with \(f\) on the small inner cube and with \(\gamma\) along the radial directions, sending \(\partial I^n\) to \(x_0\text{.}\)
One checks the following:
-
\(\gamma \cdot (f + g) \simeq \gamma \cdot f + \gamma \cdot g\text{;}\)
-
\((\gamma * \eta) \cdot f \simeq \gamma \cdot (\eta \cdot f)\text{;}\)
-
\(c_{x_1} \cdot f \simeq f\text{;}\)
-
\(\beta_\gamma\) is well-defined.
If \(x_0 = x_1\) and \(X\) is path-connected, this yields an action of the fundamental group on the higher homotopy groups:
\begin{equation*}
\pi_1(X, x_0) \times \pi_n(X, x_0) \to \pi_n(X, x_0), \qquad ([\gamma], [f]) \mapsto [\gamma \cdot f].
\end{equation*}
Definition 5.9.
A space \(X\) is called abelian if \(\pi_1\) acts trivially on \(\pi_n\) for all \(n \geq 1\text{.}\)
