Quasimorphisms and bounded cohomology of transformation groups of nonorientable surfaces

Let $N_g$ be a closed connected nonorientable surface of genus $g\ge1$, 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 $N_g$. We prove injectivity of this map for density-preserving diffeomorphisms. For $g\ge2$ and quasimorphisms vanishing on the disc braid group, we obtain an injective map to $C^0$-continuous homogeneous quasimorphisms on $Homeo_0(N_g,μ)$. We also obtain undistortion criteria in $Homeo_0(N_g)$ and $Diff_0(N_g)$. For $g=1,2$ and every $d\ge2$, we prove $$ \dim\overline H_b^d(Homeo_0(N_g,μ))=\infty,\qquad \dim\overline{EH}_b^d(Diff_0(N_g,ω))=\infty. $$ Moreover, $\overline{EH}_b^d(Homeo_0(N_2,μ))$ is infinite dimensional in every positive even degree $d$. Here $\overline{H}_b^d$ and $\overline{EH}_b^d$ denote reduced bounded cohomology and reduced exact bounded cohomology, respectively.