The Fundamental Group of the Cylinder and the Torus

# The Fundamental Group of the Cylinder and the Torus

Recall from The Fundamental Group of a Topological Product page that if $X$ and $Y$ are topological spaces then the fundamental group of the product space $X \times Y$ is isomorphic to the direct product of the fundamental groups of $X$ and $Y$, that is:

(1)
\begin{align} \quad \pi_1(X \times Y, (x, y)) \cong \pi_1(X, x) \times \pi_1(Y, y) \end{align}

We will use this result to obtain the fundamental groups of the cylinder and the torus.

 Theorem 1: The fundamental group of the unit cylinder is isomorphic to $\mathbb{Z}$.
• Proof: The unit cylinder is the topological product of $S^1$ with $I$. • The fundamental group of the unit circle $S^1$ is $\mathbb{Z}$ and the fundamental group of the closed unit interval $D^1$ is the trivial group. Therefore:
(2)
\begin{align} \quad \pi_1(S^1 \times I, (x, y)) \cong \pi_1(S^1, x) \times \pi_1(I, y) = \mathbb{Z} \times \{ [c_y] \} \cong \mathbb{Z} \quad \blacksquare \end{align}
 Theorem 2: The fundamental group of the torus is isomorphic to $\mathbb{Z} \times \mathbb{Z}$.
• Proof: The torus is the topological product of $S^1$ with $S^1$. • The fundamental group of the unit circle $S^1$ is $\mathbb{Z}$. Therefore:
(3)
\begin{align} \quad \pi_1(S^1 \times S^1, (x, y)) \cong \pi_1(S^1, x) \times \pi_1(S^1, y) = \mathbb{Z} \times \mathbb{Z} \quad \blacksquare \end{align}

## Example 1

Let $A$ and $B$ be concentric spheres. Connect $A$ and $B$ by two tubes and delete the caps to obtain the figure $X$ below. Find $\pi_1(X, x)$. Observe that $X$ is homeomorphic to the torus. By the previous theorem we have that:

(4)
\begin{align} \quad \pi_1(X, x) \cong \mathbb{Z} \times \mathbb{Z} \end{align}

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