Welcome to part four of a six-part series where we prove that the fundamental group of the circle $\pi_1(S^1)$ is isomorphic to $\mathbb{Z}$. In this post we prove that our homomorphism from $\mathbb{Z}$ to $\pi_1(S^1)$ is surjective. The proof follows that found in Hatcher's Algebraic Topology section 1.1.

Welcome to part three of a six-part series where we prove that the fundamental group of the circle $\pi_1(S^1)$ is isomorphic to $\mathbb{Z}$. In this post we prove that our map from $\mathbb{Z}$ to $\pi_1(S^1)$ is a group homomorphism. The proof follows that found in Hatcher's Algebraic Topology section 1.1.

Welcome to part two of a six-part series where we prove that the fundamental group of the circle $\pi_1(S^1)$ is isomorphic to $\mathbb{Z}$. In this post we justify a shortcut that we never actually use in the remainder of this series, so the reader is welcome to skip this post. But I've included it since, in this series, we're closely following section 1.1 of Hatcher's Algebraic Topology.

Welcome to part one of a six-part series where we prove that the fundamental group of the circle $\pi_1(S^1)$ is isomorphic to $\mathbb{Z}$. In this post we define a map from $\mathbb{Z}$ to $\pi_1(S^1)$ and make some simple observations via pictures and an animation! The proof follows that found in Hatcher's Algebraic Topology</a>, section 1.1.