Proposition 5.10.
For each \(n \geq 2\text{,}\) \(\pi_n \colon \mathbf{Top}_* \to \mathbf{Ab}\) is a functor.
Section 5.3 Functoriality
Proof.
If \(\varphi \colon (X, x_0) \to (Y, y_0)\) is a based map, define
\begin{equation*}
\varphi_* \colon \pi_n(X, x_0) \to \pi_n(Y, y_0), \qquad [f] \mapsto [\varphi \circ f].
\end{equation*}
This is a well-defined group homomorphism. It is straightforward to verify that \((\mathrm{Id}_{(X, x_0)})_* = \mathrm{Id}_{\pi_n(X, x_0)}\) and \((\varphi \circ \psi)_* = \varphi_* \circ \psi_*\text{.}\)
Corollary 5.11.
If \(\varphi \colon (X, x_0) \to (Y, y_0)\) is a based homotopy equivalence, then \(\varphi_* \colon \pi_n(X, x_0) \to \pi_n(Y, y_0)\) is an isomorphism.
Exercise 5.12.
If \(\varphi \colon X \to Y\) is a (not necessarily based) homotopy equivalence, it is still true that \(\varphi_* \colon \pi_n(X, x_0) \to \pi_n(Y, \varphi(x_0))\) is an isomorphism.
Example 5.13.
\(\pi_i(\mathbb{R}^n) = 0\) for all \(i\text{,}\) since \(\mathbb{R}^n\) is homotopy equivalent to a point.
