Signature of spin 4-manifolds and spin mapping class groups

Lisca and Parma showed that every smooth 4-manifold admits a peculiar kind of handle decomposition, which they call horizontal. As a consequence, it is possible to prove that every smooth closed 4-manifold is the union of an achiral Lefschetz fibration over $D^2$ and a handlebody bundle over $S^1$, glued along their boundaries. I use this splitting to study spin 4-manifolds, obtaining a new proof of Rokhlin’s theorem on the signature. The key technical step involves finding a presentation of the even spin mapping class group of a closed orientable surface, using the method of Hatcher-Thurston and Wajnryb. In order to compute the signature, I use results of Endo-Nagami and Kuno-Sato on Meyer’s cocycle.