# Universal cover of a space with cell attached

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 $$\pi_1(X)$$?

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

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