A map between connected $n$-dimensional CW complexes is a homotopy equivalence 
Show that a map between connected $n$-dimensional CW complexes is a homotopy equivalence if it induces an isomorphism on $\pi_i$ for $i\leq n$. [Pass to universal covers an use homology].

I'm in the middle of proving the above statement. As in hint, for the given map $f:X\to Y$ between connected $n$-dimensional CW complexes, let $\tilde{f}:\tilde{X}\to\tilde{Y}$ be a lift of $f$ where $\tilde{X}$ and $\tilde{Y}$ are universal covers of $X$ and $Y$. Using the fact that $f:\pi_i(X)\to\pi_i(Y)$ is an isomorphism for all $i\leq n$ and $\tilde{X}$ and $\tilde{Y}$ are simply connected, $\tilde{f}_*:\pi_i(\tilde{X})\to\pi_i(\tilde{Y})$ is an isomorphism for all $i\leq n$. Using relative Hurewicz theorem, one can show $\tilde{f}_*:H_i(\tilde{X})\to H_i(\tilde{Y})$ for $i\leq n$.
Now the problem is that I want to say $\tilde{X}$ and $\tilde{Y}$ have $n$-dimensional CW complex structures. But it seems to be a very nontrivial fact. Is there another way that avoids using this fact?

It is a very nontrivial fact because it appeared in a paper once? I do not agree. This fact is not hard to prove, and can be assigned as an exercise in a first algebraic topology course. So I will point you the right directions and let you give a proof, which hopefully you will write up as an answer.
Hint. Use Hatcher Proposition A.2 to define the cell structure. Clearly you want cells upstairs to project to cells downstairs, so you need to understand something about lifting along covering maps: so see the section titled "Lifting properties" in Hatcher 1.3.
