Quasimorphisms and bounded cohomology of transformation groups of nonorientable surfaces

Let Ng be a closed connected nonorientable surface of genus g≥1, let $ω$ be a smooth positive density, and let $μ$ be a good measure. In this paper we generalize the Gambaudo--Ghys construction using homogeneous quasimorphisms on the central quotients of the braid groups of Ng. We prove injectivity of this map for density-preserving diffeomorphisms. For g≥2 and quasimorphisms vanishing on the disc braid group, we obtain an injective map to C⁰-continuous homogeneous quasimorphisms on Homeo₀(Ng,μ). We also obtain undistortion criteria in Homeo₀(Ng) and Diff₀(Ng). For $g=1,2$ and every d≥2, we prove Hbᵈ(Homeo₀(Ng,μ))=∞, ̄bᵈ(Diff₀(Ng,ω))=∞. Moreover, EH̄bᵈ(Homeo₀(N₂,μ)) is infinite dimensional in every positive even degree d. Here H̄bᵈ and EH̄bᵈ denote reduced bounded cohomology and reduced exact bounded cohomology, respectively.