I mean 'the blue region has a fundamental group that is killed by the inclusion map in the surface'. I edit it right away.

Hi! It is not obvious to me that I can deform the original embedding into yours. Should it be? Also, I have the impression that at some point (when you do the turn by one third and the inverse map) you are moving from a continuous motion to precomposing the embedding by a diffeomorphism of $\Sigma$ which is not isotopic to the identity, right?

Thank you very much for all these reference I found everything I need in it :)

@TomGoodwillie Indeed, I'm not sure what is the good notion to look at here. I'd say homotopy equivalence not to add subtleties of analytic nature.

I think I am confused because I don't really understand to which underlying geometric structure you are both referring.

Yes, but couldn't it be possible that a free homotopy class would be preserved without any of its representative being preserved?

Well, I understand that a pseudo-Anosov map cannot fix a curve, my question is about the homotopy class of a curve. In which case it is not obvious to me that your argument concludes.