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Let $X$ be an $n$-dimensional CW-complex. Let $\tilde{X}$ be its universal covering space. I want to determine if $$H_i(\tilde{X},\mathbb{Z})=0,\,\,\,\,\, i\geq n+1.$$ I would like to say that $\tilde{X}$ is a CW-complex with the same dimension as $X$. In this case, I can immediately conclude using cellular homology.

Is my statement about the CW-structure on $\tilde{X}$ true?

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Yes that is always true. In fact if $E\xrightarrow{q} X$ is a covering space of $X$, where $X$ is a CW-complex, we can always find a CW-structure on $E$ such that $q$ maps cells of $E$ homeomorphically onto cells of $X$. See this post: A covering space of CW complex has an induced CW complex structure.

Womm
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