Universal cover of a space with cell attached

Question

Let $X$ be a CW complex, and suppose $W$ is obtained from $X$ by attaching an $n$-cell to X, where $n>1$. Consider the universal cover $\widetilde{X}$ of $X$. Is there a way obtaining the universal cover $\widetilde{W}$ of $W$ from $\widetilde{X}$ by attaching $n$-cells?

Let $\varphi:S^{n-1} \to X$ be the attaching map of the $n$-cell. If I attach $n$-cells to $\widetilde{X}$ via all possible lifts of $\varphi$ to $\widetilde{X}$, then it seems the resulting space is the universal cover of $W$, but how can I prove this?

Also, is there a way to express all the lifts of $\varphi$, in terms of $\pi_1(X)$?

Answer 1

Consider any group generated by elements, with a finite set of relations. It is the fundamental group of a $2$-dimensional CW complex which can be obtained by considering a bouquet of $n$ circles and attaching $2$ cells to it. The fundamental group of the resulting $2$-CW complex is $G$, if $n=1$, the universal cover of $X$ is the line and the universal cover of the associated $2$ $CW$ complex can be the $2$-sphere ($G=\mathbb{Z}/2$,...)

If $n>2$, $\pi_1(\tilde X)=\pi_1(W)$ see the reference. Let $f:S^n\rightarrow X$ be the attaching map, since $\tilde X\rightarrow X$ is a covering, we can lift $f$ to maps $f_i:S^n\rightarrow \tilde X$ where $i\in I$ the set of connected component of the inverse image of the image of the attaching maps. It defines a manifold $\tilde W$ and a canonical map $p_W:\tilde W\rightarrow W$ which is a covering, remark that $\tilde W$ is also simply connected since it is obtained by attaching $n$-cells, $n>2$ to $\tilde X$.

https://mathoverflow.net/questions/57586/on-the-fundamental-group-of-a-finite-cw-complex

Answer 2

The answer to the first question should be yes; more precisely perhaps fix a basepoint $s_0\in S^{n-1}$, then you can lift $\varphi$ once for each $c \in p^{-1}(\varphi(s_0))$ where $p:\tilde X\to X$ is the covering map.

Call that lift $\varphi_c$, and then glue an $n$-cell to $\tilde X$ along each $\varphi_c$ to get $V$ (let's call it that until we know what it is).

We get an obvious map $q:V\to W$ induced by $p$. Moreover, $\tilde X$ is simply-connected, and we added $n>1$-cells to it, so $V$ is also simply-connected. So it suffices to show that $q$ is a covering map.

Let $w\in W$ not in the new cell. Then clearly since the cell is closed, we can find an open neighbourhood that doesn't touch the cell, and so it's just like in $X$, and clearly we have a covering over $w$.

Now if $w$ is in the interior of the new cell, then this interior is open in $W$, and over it we just have one copy for each $c$

It remains to look at the case where $w$ is on the boundary of the new cell. For this one, you can take a little open set around $w$ in $X$, and a little open set around it in the cell, and lift them at the same time.

This is just a sketch, but hopefully you'll know how to fill in the details