We introduce a space of stable meromorphic differentials with poles of prescribed orders and define its tautological cohomology ring. This space, just as the space of holomorphic differentials, is stratified according to the set of multiplicities of zeros of the differential. The main goal of this paper is to compute the Poincar 'e-dual cohomology classes of all strata. We prove that all these classes are tautological and give an algorithm to compute them. In a second part of the paper we study the Picard group of the strata. We use the tools introduced in the first part to deduce several relations in these Picard groups.
* Induction formula for singularities in families. The central result of the present work is the induction formula of . A similar formula has been proved by Kazarian, Lando and Zvonkine for classes of singularities in families of genus 0 stable maps (see @cite_28 ). Their formula contains only the genus 0 part of our induction formula.
@cite_28
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We consider families of curve-to-curve maps that have no singularities except those of genus 0 stable maps and that satisfy a versality condition at each singularity. We provide a universal expression for the cohomology class Poincar 'e dual to the locus of any given singularity. Our expressions hold for any family of curve-to-curve maps satisfying the above properties.