arXiv:2101.11971v2 Announce Type: replace Abstract: Given a measurable cocycle with values in $\mathrm{Homeo}^+(\mathbb{S}^1)$, we construct an $\mathrm{L}^\infty$-parametrized Euler class in bounded cohomology. We show that such a class vanishes if and only if the cocycle can be lifted to $\mathrm{Homeo}^+_{\mathbb{Z}}(\mathbb{R})$ and it admits an equivariant family of points. We recall the notion of semicohomologous cocycles and we show that two measurable cocycles are semicohomologous if and only if they induce the same parametrized Euler class. Since for minimal cocycles, semicohomology boils down to cohomology, the parametrized Euler classes of minimal cohomologous cocycles are equal. We conclude by studying the vanishing of the real parametrized Euler class and we obtain some results about elementary cocycles.
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