By a theorem of Artin and Griffiths, every sufficiently small Zariski open of a complex variety admits the structure of an iterated punctured curve fibration. Said another way, the absolute Galois group of a complex function field admits the structure of an (inverse limit of) iterated free by free groups with monodromy given by mapping classes. In the present talk, we describe joint work in progress with Benson Farb and Mark Kisin to use the theory of surface braids to construct Galois cohomology classes and control their behavior under finite extensions with specified ramification. Time permitting, we will sketch applications and limitations of this method for understanding how hard it is to solve a general degree n polynomial.