  This paper investigates the relationship between strata of abelian
differentials and various mapping class groups afforded by means of the
topological monodromy representation. Building off of prior work of the
authors, we show that the fundamental group of a stratum surjects onto the
subgroup of the mapping class group which preserves a fixed framing of the
underlying Riemann surface, thereby giving a complete characterization of the
monodromy group. In the course of our proof we also show that these "framed
mapping class groups" are finitely generated (even though they are of infinite
index) and give explicit generating sets.
