I'm trying to get my head around covering maps and would just like some help with this (probably) rather simple problem. My attempt was to make $\mathbb{R}^2$ a universal covering of $S^1\vee S^2$.
I want to find a universal cover of f $S^1\vee S^2$. My idea was to proceed as follows: Consider $S^1$ as the unit interval quotiented by the equivalence relation $0\sim 1$. Consider the unit sphere $S^2$ as the unit square quotiented by following relation $(0,y)\sim (1,y)$ & $(x,0)\sim (y,0) \forall x,y$ & $(x,1)\sim (y,1) \forall x,y$. I.e as the unit cylinder with top and bottom sides considered as single points.
Now my idea was to join these two objects together and somehow embed them in $\mathbb{R}^2$ to get a universal covering. I'm not sure how to proceed.
I would like to know, (a) is there a way of doing this, (b) is there a better way of doing it, and (c) how we use this to find the fundamental group of $S^1\vee S^2$?
Best Answer
As the universal cover contains a subset homeomorphic to $S^2$ it cannot be embedded into $R^2$. But if you want keep the style of your quotient space consider $\mathbb R \times {0} \cup [2n,2n+1] \times [0,1]$ with the equivalence relation $\forall x,y \in [2n,2n+1] \, (x,0)\sim(y,0), (x,1)\sim(y,1)$ and $(2n,x)\sim(2n+1,x)$. This of course looks like $\mathbb R \cup_{i \in Z} S^2$ as Tamaroff suggested.