Correction terms, Z2Z2-Thurston norm, and triangulations

We explicitly construct layered-triangulations for manifolds admitting a genus-one open book decomposition with connected binding. This construction gives an upper bound to the complexity of these manifolds. In a different direction, we show that the correction terms in Heegaard Floer homology give a lower bound to the genus of one-sided Heegaard splittings and the Z2Z2-Thurston norm. Using a result of Jaco–Rubinstein–Tillmann, this gives a lower bound to the complexity of certain closed 3-manifolds. We illustrate our theorems with some examples.