Show that the isomorphism \(\pi_1(X \times Y) \approx \pi_1(X) \times \pi_1(Y)\) in Proposition 1.12 is given by \([f] \mapsto (p_{1*}([f]), p_{2*}([f]))\) where \(p_1\) and \(p_2\) are the projections of \(X \times Y\) onto its two factors.
Solution
Fix the basepoint \((x_0,y_0)\in X\times Y\) and define \[ \Psi:\pi_1(X\times Y,(x_0,y_0)) \longrightarrow \pi_1(X,x_0)\times\pi_1(Y,y_0) \] by \[ \Psi([f])=([p_1\circ f],[p_2\circ f]). \]
1. The map \(\Psi\) is a homomorphism.
For based loops \(f,g\) in \(X\times Y\), \[ p_i\circ(f\cdot g)=(p_i\circ f)\cdot(p_i\circ g) \] up to the standard concatenation parametrization. Hence \[ \Psi([f][g]) = \bigl([p_1f][p_1g],[p_2f][p_2g]\bigr) = \Psi([f])\Psi([g]). \] The definition depends only on the based homotopy class because composing a based homotopy with either projection gives a based homotopy in the corresponding factor.
2. Define\[ \Theta:\pi_1(X,x_0)\times\pi_1(Y,y_0) \longrightarrow \pi_1(X\times Y,(x_0,y_0)) \]by\[ \Theta([a],[b])=[t\mapsto(a(t),b(t))]. \]Then \(\Theta\) is well defined.
If \(a\simeq a'\) and \(b\simeq b'\) relative to endpoints, let \[ A:I\times I\to X, \qquad B:I\times I\to Y \] be the two based homotopies. Then \[ H(s,t)=(A(s,t),B(s,t)) \] is a based homotopy in \(X\times Y\) from \((a,b)\) to \((a',b')\). Thus the class defining \(\Theta\) depends only on \([a]\) and \([b]\).
3. One has\[ \Psi\circ\Theta=\operatorname{id}. \]
For \(([a],[b])\), \[ \Psi\Theta([a],[b]) = \Psi([t\mapsto(a(t),b(t))]) = ([a],[b]), \] since the two projections recover \(a\) and \(b\) exactly.
4. One also has\[ \Theta\circ\Psi=\operatorname{id}. \]
If \(f:I\to X\times Y\) is a based loop, then for every \(t\) \[ f(t)=\bigl(p_1(f(t)),p_2(f(t))\bigr). \] Therefore the loop used to define \[ \Theta\Psi([f]) \] is literally \(f\), so \[ \Theta\Psi([f])=[f]. \]
5. Thus \(\Psi\) is the product isomorphism, and it is exactly\[ [f]\longmapsto\bigl(p_{1*}([f]),p_{2*}([f])\bigr). \]
By <1>3–<1>4, \(\Theta\) is a two-sided inverse of \(\Psi\). The displayed formula is the definition of the two induced projection maps.