Surgery on 3-manifolds with \(\mathbb{S}^ 1\)-actions (Q1919278)

The authors of the paper study the topological structure of all 3-manifolds obtained by surgery along principal fibers of a closed orientable manifold equipped with an action of the circle \(S^1\). They make use of the well-known equivariant classification of 3-manifolds with \(S^1\)-actions due to \textit{F. Raymond} [Trans. Am. Math. Soc. 131, 51-78 (1968; Zbl 0157.30602)]. Using their methods, the authors provide new proofs of the classical results by \textit{W. Heil} [Proc. Am. Math. Soc. 37, 609-614 (1973; Zbl 0249.57004); Yokohama Math. J. 22, 135-139 (1974; Zbl 0297.57006)], as well as by \textit{L. E. Moser} [Pac. J. Math. 38, 737-745 (1971; Zbl 0202.54701)]. For the manifolds considered in the paper, they completely specify the Seifert invariants. As applications of the results, they classify the closed 3-manifolds with \(S^1\)-actions obtained by Dehn surgeries along certain Seifert links in the 3-sphere \(S^3\). Moreover, they find geometric presentations of the fundamental groups of the manifolds in question which arise from Heegaard diagrams.