CW structure of the universal cover

Consider the universal covering $$p:\tilde X\to X$$. Given a cw structure on $$X$$, I want to understand how to find the cw structure of $$\tilde X$$.

Let's take an example $$p:\mathbb R\to S^1;\;\;t\mapsto e^{i2\pi t}$$

We have the usual CW structure on $$S^1$$ consisting on having one $$0$$-cell $$e^0$$ and one $$1$$-cell $$e^1$$. How to get CW structure of $$\mathbb R$$ from that of $$S^1$$ ? I know that real line admits the structure of $$1$$-dimensional CW- complex with the integers as zero-cells and the intervals $$[n, n + 1]$$ as 1-cells but I want the structure coming from the structure of $$S^1$$ by the universal cover map. Thank you for your help!

When you have a covering map $$p:\tilde{X}\to X$$ with a CW structure on $$X$$, you can use the homotopy lifting property to lift each cell of $$X$$ to a collection of cells of $$\tilde{X}$$, since you can think of maps $$D^k\to X$$ as being homotopies if you parameterize things right. In particular, given a cell $$e^k:D^k\to X$$, there is a map $$\tilde{e}^k:D^k\to \tilde{X}$$ called a lift such that $$p\circ \tilde{e}^k=e^k$$. Lifts are not unique.

Let's say $$p$$ is a universal covering space for simplicity.

The $$0$$-cells in $$X$$ each lift to a collection of $$0$$-cells in $$\tilde{X}$$ in correspondence to $$\pi_1(X)$$. It's easier to keep track of things when $$X$$ has a single $$0$$-cell that is the basepoint: then the $$0$$-cells of $$\tilde{X}$$ are in correspondence with $$\pi_1(X,*)$$. Since $$0$$-cells are just points, lifts are points from the inverse image $$p^{-1}(*)$$. Let's say $$X$$ has a single $$0$$-cell for simplicity.

The $$1$$-cells in $$X$$ are then loops (since the boundaries are attached to the single $$0$$-cell $$*$$), and so they can be thought of as elements of $$\pi_1(X,*)$$. These lift to paths between lifts of the basepoint. In particular, if a $$1$$-cell $$e^1$$ corresponds to $$a\in\pi_1(X,*)$$, and if $$*_x\in\tilde{X}$$ is a lift of the basepoint corresponding to $$x\in\pi_1(X,*)$$, then there is a lift of $$e^1$$ that is a path from $$*_x$$ to $$*_{xa}$$. This is essentially reiterating part of the construction of the universal covering space.

The $$2$$-cells and higher are simply connected, and so there is one lift of each per lift of the basepoint. It is a little tricky to figure out what happens to the attachment map, but usually you can see the order of the $$1$$-cells along the boundary, then follow the lifts of those cells in the cover.

In your case of $$S^1$$ with one $$0$$-cell and one $$1$$-cell, the lifts of the $$0$$-cell are $$\mathbb{Z}\subset\mathbb{R}$$. The loop lifts to paths from $$n$$ to $$n+1$$. So, the CW structure of $$\mathbb{R}$$ that you describe is the one coming from the universal covering map $$t\mapsto e^{2\pi i t}$$.

• what do you mean by lifting a cell ? do you mean $p^{-1}(e_i)$ is a collection of cells $\tilde e_i$ in $\tilde X$ one on each sheet ? Is the dimension of the obtained cells equal to the dimension of the original lifted cell $e_i$ in $X$? Isn't a one cell just a segment, why you say it is a loop as if it is a circle which in this case the whole skeleton $X^1$ obtained by attaching the one 1-cell to the 0-cell and so it is not the 1-cell.. finally why the 2-cells and higher are contractible in $X$? – palio Oct 22 '18 at 21:25
• @palio I added some more explanation of what a lift is (though it might be worth looking up the "homotopy lifting property" in some reference), and lifts are not inverse images in general. And, why I called the cell a "loop" is that the image of the $1$-cell $e^1:D^1\to X$ is a loop if there is only a single $0$-cell. It might be more precise to say that the closure of the open cell is a loop. – Kyle Miller Oct 22 '18 at 22:41