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