Below is a general method —a recipe, if you will —for computing the universal cover of the wedge sum $X\vee Y$ of arbitrary topological spaces $X$ and $Y$. This is simply a short-and-quick guideline that my prof mentioned in class, and I thought it'd be helpful to share on the blog. To help illustrate each step, we'll consider the case when $X=T^2$ is the torus and $Y=S^1$ is the circle.       

Welcome to part five 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 injective. The proof follows that found in Hatcher's Algebraic Topology section 1.1.

Have you ever read Tristan Needham’s Visual Complex Analysis? I highly recommend this book as a supplement to a standard undergrad/grad course in complex analysis. It's nothing (nothing!) like your usual textbook. The author writes to build your intuition and insight, so it's warm like a conversation and not cold like some math texts. It’s also loaded with illustrations (hence the title), historical background, and context. For example, did you know