An alternative way to define an isomorphism for the fundamental group of a product space.
by Hidenori
Proposition
Show that the isomorphism $\pi_1(X \times Y) \approx \pi_1(X) \times \pi_1(Y)$ in this proof 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
In the proof, we defined $\phi: \pi_1(X \times Y, (x_0, y_0)) \rightarrow \pi_1(X, x_0) \times \pi_1(Y, y_0)$ such that $\phi([(g, h)]) = [g] \times [h]$ for all $[(g, h)] \in \pi_1(X \times Y)$.
\[\begin{align*} (p_{1*}([(g, h)]), p_{2*}([(g, h)])) &= ([p_1(g, h)], [p_2(g, h)]) \\ &= ([g], [h]) \\ &= [g] \times [h] = \phi([(g, h)]). \end{align*}\]Subscribe via RSS