In the context of the absolute Galois group action on the fundamental group of moduli spaces of curves, the goal of the Grothendieck-Teichmüller theory is to give explicit and combinatorial results. In this talk, we will explain how to define a Galois action by pants decompositions on surfaces and how this theory, combined with mapping class groups representation by braids and Serre's cohomology theory, leads to a result for the Galois action on profinite stack inertia.