Birkhoff sections of Anosov flows

An Anosov flow is a globally hyperbolic dynamical system which can be studied in dimension 3 using some transverse surfaces called Birkhoff section. There exist three families of Birkhoff sections which can not co-exist, and they help understand the nature of the flow. I will explain a relation between the existence of these Birkhoff sections with the topology of the orbit space of the flow.