Constructing a simply connected covering space

Question

"Construct a simply connected covering space of the space that is the union of the sphere S2 with two of its intersecting diameters." can anyone help me with this? i don't know how to think , all propositions i learned were so general that i really don't know where to use them in the construction.I read a proof for hatcher exercice 4 page 79, that is some how similar to my question , but still i can't solve mine.

Answer 1

Here, basically I am going to use the idea given by HATCHER (page $65$) i.e suppose $X$ is union of subspace of $A$ and $B$ for which simply connected covering spaces are already known. Then, how one can attempt to build a simple connected covering space of $X$.

Let's do step by step.

$(1)$ Let $X$ be $\Bbb S^2$ with a diameter. Then, the universal cover would be a infinite chain of sphere $\{...,X_{-1},X_0,X_1,...\}$ and north pole of $X_i$ attached with south pole of $X_{i+1}$ by an edge. The covering map would be $X_n$ maps to $\Bbb S^2$ for each $n\in\Bbb Z$ and edges will map to the diameter with proper orientation. Named this universal cover of $X$ as $Y$.

$(2)$ Now, for the case $\Bbb S^2$ with two intersecting diameters we proceed in following ways.

Consider a copy of $Y$ lying horizontally. This is our $Y_1$ i.e. $Y_1=Y$.

Now, with each midpoint of edges of $Y$ attach a copy of $Y$ vertically by gluing the midpoint of the edge connecting $X_0$ and $X_1$. This space is $Y_2$.

Now, again with each midpoint of edges of vertically lying copies of $Y_2 -Y_1$ attach a copies of $Y$ horizontally like previous way to construct $Y_3$.

Now, keep on doing this operation.

Claim : $\displaystyle\overline Y:= \bigcup_{n \geq 1} Y_i$ is the universal cover of $\Bbb S^2$ with two intersecting diameters. In this case, the covering map would be following. Each copy of sphere maps to the original sphere. Every horizontal edges will map to the horizontal diameter and every vertical edges will map to the vertical diameter.