A Recipe for the Universal Cover of X⋁Y

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.

The Ingredients

This is sort of redundant, but since I'm calling today's post a "recipe," I might as well list the ingredients!
  • two topological spaces $X$ and $Y$
  • the universal cover $\tilde{X}$ of $X$
  • the universal cover $\tilde{Y}$ of $Y$

The Instructions

Step 1:

Start with the universal cover $\tilde{X}$ of $X$ and let $p\in X$ and $q\in Y$ be the points that are identified ($p\sim q$) in the wedge $X\vee Y$.
 
 

Step 2:

To each lift $\tilde{p}\in\tilde{X}$ of $p$, attach one copy of the universal cover $\tilde{Y}$ of $Y$ by identifying that $\tilde{p}$ to some lift $\tilde{q}$ of $q$.
 
 

Step 3:

To each unidentified $\tilde{q}$ (in each copy of $\tilde{Y}$) attach a copy of $\tilde{X}$ by identifying that $\tilde{q}$ to some lift $\tilde{p}\in \tilde{X}$.
 
 

Step 4:

On each copy of $\tilde{X}$ from step 3, repeat step 2 - pictured below - then repeat step 3, then repeat step 2, then repeat step 3, then.... The space you obtain after this infinite process is the universal cover of $X\vee Y$!