First Fundamental Group of 1-Sphere

Theorem

Let $\mathbb S^1$ be the $1$-sphere.

Let $\struct {\map {\pi _1} {\mathbb S^1}, \ast}$ be the first fundamental group of $\mathbb S^1$.

Let $\struct {\Z, +}$ be the additive group of integers.

Then $\struct {\map {\pi _1} {\mathbb S^1}, \ast}$ is isomorphic to $\struct {\Z, +}$.

Proof

We are given that $\struct {\map {\pi _1} {\mathbb S^1}, \ast}$ is the first fundamental group of $\mathbb S^1$.

Let $x_0 \in \mathbb S^1$.

Let $\struct{\mathbb S^1, x_0}$ be the pointed topological space for $\mathbb S^1$.

Since Fundamental Group is Independent of Base Point for Path-Connected Space, what to be proved is:

$\struct {\map {\pi _1} {\mathbb S^1, x_0}, \ast} \cong \struct {\Z, +}$

where the map:

$\ast : \map {\pi_1} {\mathbb S^1, x_0} \times \map {\pi_1} {\mathbb S^1, x_0} \to \map {\pi_n} {\mathbb S^1, x_0}$

denotes the concatenation of homotopy classes of paths.

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Hence $\struct {\map {\pi _1} {\mathbb S^1}, \ast}$ is isomorphic to $\struct {\Z, +}$.