Classification and equivariant cohomology of circle actions on 3d manifolds

TL;DR

This paper extends the classification of circle actions on 3d manifolds to include those with boundaries, introduces new parameters, and analyzes their equivariant cohomology, Betti numbers, and formality properties.

Contribution

It generalizes existing classifications to manifolds with boundaries by adding numeric parameters and cycle graphs, and provides a detailed analysis of their equivariant cohomology.

Findings

Classification extended to manifolds with boundaries

Explicit computation of equivariant Betti numbers and Poincaré series

Discussion of equivariant formality properties

Abstract

The classification of Seifert manifolds was given in terms of numeric data by Seifert in 1933, and then generalized by Orlik and Raymond in 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 union of cycle graphs. Then we describe the equivariant cohomology of 3d manifolds with circle actions in terms of ring, module and vector-space structures. We also compute equivariant Betti numbers and Poincar\'{e} series for these manifolds and discuss the equivariant formality