Classification and equivariant cohomology of circle actions on 3d manifolds

The classification of Seifert manifolds was given in terms of numeric data by Seifert (1933), and then generalized by Raymond (1968) and Orlik and Raymond (1968) to circle actions on closed 3d manifolds. In this paper, we further generalize the classification to circle actions on 3d manifolds with boundaries by adding a numeric parameter and a graph of cycles. Then, we describe the rational equivariant cohomology of 3d manifolds with circle actions in terms of ring, module and vector-space structures. We also compute equivariant Betti numbers and Poincaré series for these manifolds and discuss the equivariant formality.