Relative homological representations of framed mapping class groups

Let $\mathrm{\Sigma }$ be a surface with either boundary or marked points, equipped with an arbitrary framing. In this note we determine the action of the associated “framed mapping class group” on the homology of $\mathrm{\Sigma }$ relative to its boundary (respectively marked points), describing the image as the kernel of a certain crossed homomorphism related to classical spin structures. Applying recent work of the authors, we use this to describe the monodromy action of the orbifold fundamental group of a stratum of abelian differentials on the relative periods.