I am studying Hatcher's Algebraic topology. I am reading 3G about Transfer homomorphism. But most of the results are deemed obvious and I don't understnad why.

"Let $\pi:\tilde X\to X$ be an $n$-sheeted covering space for some finite $n$. Then there are $n$ distinct lifts $\tilde \sigma : \Delta^n\to \tilde X$ of a map $\sigma: \Delta^n\to X$."

I don't understand this part. So if $n\geq 3$ then the $n$-simplex is just the $S^{n-1}$, which has trivial fundamental group, and hence A lift exists, that's all I got. If $n=2$, then the fundamental group is $\mathbb{Z}$, so I don't even know why a lift exists. Can you explain this? I omit some parts of the books so I think I miss the details somewhere.

Also, there are more parts I don't understnad, can you recommend me a book about the details for this part of Hatcher's book?