4
$\begingroup$

For any manifolds $M$, a homotopy class of diffeomorphism gives rise to an automorphism of $\pi_1(M)$ (up to conjugacy since we are dealing with free homotopies). Moreover, in the specific case of surfaces, Dehn-Nielsen-Baer's theorem tells us that the map $\operatorname{Mod}(M) \to \operatorname{Out}(\pi_1(M))$ is an isomorphism. This is actually really specific to surfaces and do not hold in higher dimensions in general. However, I am wondering if this theorem could be generalized to topological branched cover of a given degree.

This indeed works in the case of a branched cover of the sphere as the homotopy type of the map is determined by its degree.

So more generally, if $f,g : \Sigma_h \to \Sigma_g$ are a topological branched cover of surfaces such that the induced maps on the fundamental groups are the same up to conjugacy, then is $f \simeq g$? If so, is there a good reference where I could read about it? If not, is there an easy counter example?

I was thinking that a possible counter example might come from precomposing $f$ with a Dehn twist $m$ around an essential curve $\gamma$ such that $f(\gamma) \simeq \ast,$ but I haven't work out this example in detail yet.

Edit: It may be worth mentioning that the homotopies that I am considering are not relative to the branched points. In particular, the branching data is not necessarily fixed by the homotopy class.

$\endgroup$
4
$\begingroup$

This is true in general, since $\Sigma_g$ is a $K(\pi,1)$ for $g\ge 1$, and if $A$ and $B$ are groups then the set of unbased homotopy classes $[K(A,1),K(B,1)]$ is in one-to-one correspondence with homomorphisms from $A$ to $B$ modulo conjugacy in $B$. This latter fact is well-known, and follows from Proposition 4A.2 of Hatcher's textbook.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.