Constructions of open book decompositions

We introduce the naive notion of a relative open book decomposition for contact 3-manifolds with torus boundary. We then use this to construct nice, minimal genus open book decompositions compatible with all of the universally tight contact structures (as well as a few others) on torus-bundles over S¹, following Honda's classification. In an accurate sense, we find Stein fillings of 'half' of the torus bundles. In addition, these give the first examples of open books compatible with the universally tight contact structures on circle bundles over higher genus surfaces, as well, following a pattern introduced by a branched covering of B⁴. Some interesting examples of open books without positive monodromy are emphasized.