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 $t\mapsto e^{2\pi i t}$.

  • $\begingroup$ 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 $X$? $\endgroup$ – palio Oct 22 '18 at 21:25
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    $\begingroup$ @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. $\endgroup$ – Kyle Miller Oct 22 '18 at 22:41

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