[Math] Universal covering map and induced maps on homology and homotopy

Question

I had an additional question regarding universal covering maps. If $p:U\rightarrow X$ is a universal covering map for space $X$ does it induce isomorphisms on homology $H_i$ for $i>1$. Or if that's not the case does it induce isomorphisms on homotopy groups $\pi_i$ for $i>1$. If this is not true does it work if we assume $X$ is a CW complex. I apologize if these are trivial facts, I'm just trying to fill some gaps in my knowledge.

Answer 1

No. For example, $T^2$ has contractible universal cover $\mathbb{R}^2$, but $H_2(T^2) = \mathbb{Z}$.