The Fundamental Group of a Topological Product

# The Fundamental Group of a Topological Product

 Theorem 1: Let $X$ and $Y$ be topological spaces and let $X \times Y$ denote the product of these spaces. Then $\pi_1(X \times Y, (x, y))$ is isomorphic to $\pi_1(X, x) \times \pi_1(Y, y)$.
• Proof: Let $p_1 : X \times Y \to X$ and $p_2 : X \times Y \to Y$ be the projection functions defined for all $(x, y) \in X \times Y$ by:
(1)
\begin{align} \quad p_1(x, y) = x \end{align}
(2)
\begin{align} \quad p_2(x, y) = y \end{align}
• Then $p_1$ and $p_2$ are continuous functions. Define a function $\phi : \pi_1(X \times Y, (x, y)) \to p_1(X, x) \times p_2(Y, y)$ for all homotopy classes $[\alpha] \in pi_1(X \times Y, (x, y))$ by:
(3)
\begin{align} \quad \phi ([\alpha]) = (p_{1*}([\alpha]), p_{2*}([\alpha])) \end{align}
• We first show that $\phi$ is a homomorphism. Let $[\alpha], [\beta] \in \pi_1(X \times Y, (x, y))$. Then:
(4)
\begin{align} \quad \phi ([\alpha][\beta]) &= \phi([\alpha\beta]) \\ &= (p_{1*}([\alpha\beta]), p_{2*}([\alpha\beta]) \\ &= ([p_1 \circ \alpha\beta], [p_2 \circ \alpha\beta]) \\ &= ([(p_1 \circ \alpha)(p_1 \circ \beta)], [(p_2 \circ \alpha)(p_2 \circ \beta)]) \\ &= ([p_1 \circ \alpha][p_1 \circ \beta], [p_2 \circ \alpha][p_2 \circ \beta]) \\ &= ([p_1 \circ \alpha], [p_2 \circ \alpha]) ([p_1 \circ \beta], [p_2 \circ \beta]) \\ &= \phi([\alpha]) \phi([\beta]) \end{align}
• Furthermore, $\phi$ is bijective since its range is composed of the induced mappings of projections. So $\phi$ is an isomorphism and:
(5)
\begin{align} \quad \pi_1(X \times Y, (x, y)) \cong \pi_1(X, x) \times \pi_1(Y, y) \quad \blacksquare \end{align}