Universal covers of CW complexes and $n$-skeletons

Question

I was wondering if the following were true. If $U$ is the universal cover of a CW complex $X$, then can we conclude that $U^n$ is the universal cover for the $n$-skeleton $X^n$? It seems like its valid, but I'm not sure. I haven't had a formal study on $CW$ complexes so there is still some gaps in my understanding of these and similar results. Also is it also true that the universal cover of an $n$-dimensional CW complex is also $n$-dimensional? I would appreciate any feedback. Thanks.

Answer 1

This is true if $n \ge 2$. In that case, the inclusion $X^{(n)} \hookrightarrow X$ is an isomorphism on fundamental group, and hence by lifting lemmas what you say holds true.

Answer 2

This is not true. Consider $X = S^2$ with a single $0$-cell, a single $1$-cell and two $2$-cells (the hemispheres). As $X$ is simply connected, it is its own universal cover, so $U = X$. However, $X^1 = S^1$ which does not have universal cover $U^1 = S^1$ (the universal cover of $S^1$ is $\mathbb{R}$).