📂Topological Data Analysis

The Fundamental group of a Product Space is Isomorphic to the Product of the Fundamental groups

Theorem

Let $X, Y$ be a topological space. The fundamental group of the (topology) Cartesian product is isomorphic to the Cartesian product of the groups of each respective component. $$ \pi_{1} \left( X \times Y , \left( x_{0} , y_{0} \right) \right) \simeq \pi_{1} \left( X, x_{0} \right) \times \pi_{1} \left( Y, y_{0} \right) $$ In particular, if $X, Y$ are all path-connected, then one can omit the base point as follows: $$ \pi_{1} \left( X \times Y \right) \simeq \pi_{1} \left( X \right) \times \pi_{1} \left( Y \right) $$

Proof ¹

For the Cartesian product of topological spaces, the following two properties are equivalent.

• $f : Z \to X \times Y$ is continuous.
• $f(z) = \left( g(z) , h(z) \right)$ in which both $g : Z \to X$ and $h(z) : Z \to Y$ are continuous.

Therefore, to say $f$ takes $X \times Y$ as a base point is equivalent to saying the two paths $g, h$ each take $x_{0} \in X$ and $y_{0} \in Y$ as their respective base points. This argument also applies to homotopies $F : I^{2} \to Z$, $G : I^{2} \to X$, and $H: I^{2} \to Y$. Therefore, $$ \phi : \pi_{1} \left( X \times Y , \left( x_{0} , y_{0} \right) \right) \to \pi_{1} \left( X, x_{0} \right) \times \pi_{1} \left( Y, y_{0} \right) $$ if we define as $\phi : [f] \mapsto \left( [g] , [h] \right)$, this is trivially a group homomorphism, and since $\phi$ is bijective, it also becomes an isomorphism.

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