I am reading about the Birman-Hilden conjecture, specifically the thesis of Rebbeca Winarski and there is some basic point I am not getting.

We consider a covering space $p: Y \rightarrow X$ between two connected oriented surfaces. Let $\mathrm{Mod}(X)$ be the mapping class group of $X$ and $\mathrm{LMod}(X)$ be the subgroup of $\mathrm{Mod}(X)$ of the isotopy classes of diffeomorphisms that lift to diffeomorphisms of $Y$.

$\mathrm{LMod}(X)$ must be a finite index subgroup but I do not see why.


So, this is what I found.

As $p$ is a covering map between surfaces, it is finite. Suppose that $p$ is a $n$-fold covering map.

Choose a point $x\in X$ and choose $y\in p^{-1}(x)$. Then $p_*(\pi_1(Y,y))$ has index $n$ in $\pi_1(X,x)$.

Define $M$ to be the set of subgroups having index $n$ in $\pi_1(X,x)$. This set is finite because $\pi_1(X,x)$ is finitely generated.

Let $\mathrm{Mod(X,x)}$ be the group of isotopy classes of diffeomorphisms fixing $x$ and $\mathrm{LMod(X,x)}$ be the subgroup of $\mathrm{Mod(X,x)}$ that whose elements lift to elements of $\mathrm{Mod(Y,y)}$.

$\mathrm{Mod(X,x)}$ acts on $M$ by permuting its elements, so there is an homomorphism $\varphi$ from $\mathrm{Mod(X,x)}$ to $\Sigma_M$ (the permutation group on the elements of $M$). As $\Sigma_M$ is finite, the kernel of $\varphi$ has finite index in $\mathrm{Mod(X,x)}$.

Now, $\mathrm{LMod(X,x)}$ is the stabiliser of $p_*(\pi_1(Y,y))$ in $\pi_1(X,x)$, so it contains the kernel of $\varphi$. This proves that $\mathrm{LMod(X,x)}$ has finite index in $\mathrm{Mod(X,x)}$.

The rest follows by applying the forgetful map $\mathrm{Mod(X,x)} \rightarrow\mathrm{Mod(X)}$, which defines an isomorphism.


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