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?
Best Answer
All simplices are contractible, as was pointed out, so lifts exist.
Now given a based covering $(\tilde X,x)\to (X,b)$ and a based space $(Y,y)$ together with a map $(Y,y)\to (X,b)$ (the notation means it's a based map) satisfying the requirements for a lift to exist, then a based lift is unique.
So if the covering is $n$-sheeted, there are as many unbased lifts as there are possible choices of basepoints $x\in \pi^{-1}(b)$, that is, $n$ : there is a unique based lift for each such $x$.