  For g>2 we study the cohomology classes in the closure of a stratum of
abelian differentials defined by the boundary strata of codimension one. As an
application, we find an explicit stratification of the spin moduli space for an
odd spin structure consisting of g-1 strata D_j of codimension j-1 such that
D_j does not contain a complete subvariety for all j. We also recover some
results of Koroktin and Zograf and of Chen using a unified topological
approach.
