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}

[[/math]]