Introduction 1. The Kontsevich-Zorich cocycle 2. Variational formulas 3. A lower bound for the second Lyapunov exponent 4. The determinant locus 5. The Kontsevich-Zorich formula revisited and other formulas for the Lyapunov exponents 6. Basic currents for measured foliations 6.1. Basic currents and invariant distributions 6.2. Weighted Sobolev spaces of currents 6.3. A geometric estimate of the...

Let H be a cocommutative weak Hopf algebra and let (B,φB) left H-module algebra. In this paper, for twisted convolution invertible morphism σ:H2→B we define its obstruction θσ as Sweedler 3-cocycle with values in the center of B. We obtain that class vanish third cohomology group HφZ(B)3(H,Z(B)) if, only there exists 2-cocycle α:H2→B such H⊗B can endowed crossed product structure α keeping coho...