Cohomology classes represented by measured foliations, and Mahler's question for interval exchanges

A translation structure on (S, Σ) gives rise to two transverse measured foli- ations F, G on S with singularities in Σ, and by integration, to a pair of relative cohomology classes [F]; [G] ε H ¹ (S, Σ,R). Given a measured foliation F, we characterize the set of cohomology classes b for which there is a measured foliation G as above with b = [G]. This extends previous results of Thurston [19] and Sullivan [18]. We apply this to two problems: unique ergodicity of interval exchanges and flows on the moduli space of translation surfaces. For a fixed permutation σ ε Sd, the space ℝ ₊^d parametrizes the interval exchanges on d intervals with permutation ℓ.We describe lines ℓ in ℝ ₊^d such that almost every point in ℓ is uniquely ergodic.We also show that for σ(i) = d+1+i, for almost every s > 0, the interval exchange transformation corresponding to σ and (s, s2, : : :, sd) is uniquely ergodic. As another application we show that when k = |Σ| ≥ 2; the operation of "moving the singularities horizontally" is globally well-defined.We prove that there is a well-defined action of the groupB×ℝk-1 on the set of translation surfaces of type (S, Σ) without horizontal saddle connections. Here B ⊂ SL(2, ℝ) is the subgroup of upper triangular matrices.