Circularly ordering direct products and the obstruction to left-orderability

Motivated by the recent result that left-orderability of a group $G$ is intimately connected to circular orderability direct products $G \times \mathbb{Z}/n\mathbb{Z}$, we provide necessary and sufficient cohomological conditions such product be circularly orderable. As consequence main theorem, arrive at new characterization for fundamental rational homology 3-sphere left-orderable. Our results imply mapping class groups once-punctured surfaces, other whose actions on $S^1$ are cohomologically rigid, \mathbb{Z}/n\mathbb{Z}$ seldom We also address in general, dealing with cases factor admitting bi-invariant ordering, iterated amenable.