📂Topological Data Analysis

The Fundamental group of a Torus is Isomorphic to the Product of Two Integer groups

Theorem ¹

$$ \pi_{1} \left( T^{2} \right) \simeq \mathbb{Z} \times \mathbb{Z} $$ The fundamental group of a torus $T^{2}$ is $\mathbb{Z} \times \mathbb{Z}$.

Proof

Properties of induced homomorphisms:

• [2]: If $\varphi : X \to Y$ is a homeomorphism, then $\varphi_{\ast} : \pi_{1} \left( X, x \right) \to \pi_{1} \left( Y, \varphi (x) \right)$ is an isomorphism.

Fundamental group of a product space: $$ \pi_{1} \left( X \times Y \right) \simeq \pi_{1} \left( X \right) \times \pi_{1} \left( Y \right) $$

Fundamental group of a circle: $$ \pi_{1} \left( S^{1}, 1 \right) \simeq \mathbb{Z} $$

Since the torus $T^{2}$ is homeomorphic to $S^{1} \times S^{1}$, and the fundamental group of the circle $S^{1}$ is isomorphic to $\mathbb{Z}$, we obtain the following. $$ \begin{align*} \pi_{1} \left( T^{2} \right) \simeq & \pi_{1} \left( S^{1} \times S^{1} \right) \\ \simeq & \pi_{1} \left( S^{1} \right) \times \pi_{1} \left( S^{1} \right) \\ \simeq & \mathbb{Z} \times \mathbb{Z} \end{align*} $$

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