Show that if \(\varphi: X \to Y\) is a homotopy equivalence, then the induced homomorphisms \(\varphi_*: \pi_n(X, x_0) \to \pi_n(Y, \varphi(x_0))\) are isomorphisms for all \(n\).
Solution
Let \(\varphi:X\to Y\) be a homotopy equivalence, with homotopy inverse \(\psi:Y\to X\). Choose a path \(\eta\) from \(x_0\) to \(\psi\varphi(x_0)\) supplied by the homotopy \(\psi\varphi\simeq\operatorname{id}_X\). Naturality of homotopy groups under a homotopy gives \[ \beta_\eta\circ(\psi\varphi)_*=\operatorname{id}_{\pi_n(X,x_0)}, \] where \(\beta_\eta\) is the appropriate change-of-basepoint isomorphism. Similarly, from \(\varphi\psi\simeq\operatorname{id}_Y\) one gets a change-of-basepoint isomorphism making \(\varphi_*\psi_*\) the identity.
Thus \(\varphi_*\) has a two-sided inverse after the canonical basepoint changes. In particular \[ \boxed{\varphi_*:\pi_n(X,x_0)\xrightarrow{\cong}\pi_n(Y,\varphi(x_0))} \] for every \(n\ge1\). (For \(n=0\), the same argument says that a homotopy equivalence induces a bijection on path components.)