Let $X$ be an $n$-dimensional cell complex.

We attach an $(n+1)$-cell $e^{n+1}$ to $X$ and obtain a new cell complex $X'$.

Take the universal cover (or a general covering space) $\tilde X'$ of $X'$.

We have a covering map $\pi': \tilde X'\longrightarrow X'$.

................

Question. Whether or not can we take the $n$-skeleton of $\pi'$ (restriction of the map to the $n$-skeleton of the CW complexes) and give a new covering map $\pi: \tilde X\longrightarrow X$?

Thanks for guidance.

Let $X$ be an $n$-dimensional cell complex.

We attach an $(n+1)$-cell $e^{n+1}$ to $X$ and obtain a new cell complex $X'$.

Take the universal cover (or a general covering space) $\tilde X'$ of $X'$.

We have a covering map $\pi': \tilde X'\longrightarrow X'$.

................

Question. Whether or not can we take the $n$-skeleton of $\pi'$ (restrict the map $\pi'$ to the $n$-skeleton of the CW complex $\tilde X'$) and give a new covering map $\pi: \tilde X\longrightarrow X$?

Thanks for guidance.