2
$\begingroup$

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 $\pi_1(X)$?

$\endgroup$
1
$\begingroup$

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

| cite | improve this answer | |
$\endgroup$
1
$\begingroup$

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

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.