On mapping cones of Seifert fibered surgeries (Q2892845)

This paper addresses the question of which knots in \(S^3\) admit Seifert fibered surgeries. The author considers all the Seifert fibered spaces which are rational homology spheres, and he demonstrates that both the torsion coefficients of \(K\) and the Heegaard Floer homology \(\widehat{HFL}(K)\) offer obstructions to such a surgery. The results in this paper generalize previous work of \textit{P. S. Ozsváth} and \textit{Z. Szabó} [Geometry and Topology Monographs 7, 181--203 (2004; Zbl 1098.57019)] and \textit{P. Kronheimer} et al., [Ann. Math. (2) 165, No. 2, 457--546 (200; Zbl 1204.57038)].NEWLINENEWLINEIf rational surgery on \(K\subset S^3\) is a positive Seifert fibered space, then the author shows (1) that the torsion coefficients of \(K\) are non-negative, and (2) that certain Heegaard Floer knot groups are trivial. Analogous results hold for negative Seifert fibered spaces, where the sign refers to the orientation of the three-manifold as the boundary of the four manifold obtained by plumbing two-spheres. Wu also proves that if both the four-ball genus of \(K\) and the degree of its Alexander polynomial \(\Delta_K\) are strictly less than the genus of \(K\), then \(K\) does not admit a Seifert fibered surgery. Finally, he shows that no slice knot with both positive and negative torsion coefficients can admit a Seifert fibered surgery.NEWLINENEWLINEThe major technical tool in the paper is the rational surgery formula of Ozsváth and Szabó, which gives an isomorphism between the Heegaard Floer homology of the manifold \(Y_{\frac{p}{q}}(K)\) and the homology of a mapping cone associated to the knot Floer homology \(CFK^\infty(Y,K)\), cf. \textit{P. S. Ozsváth} and \textit{Z. Szabó} [Algebr. Geom. Topol. 11, No. 1, 1--68 (2011; Zbl 1226.57044)]. Wu's results follow from a careful analysis of the mapping cone, with particular attention paid to the \(\mathbb{Z}\slash 2\mathbb{Z}\) grading and the decomposition \(HFK^\infty(Y,K)\cong \mathcal{T}^k_{\frac{p}{q}}\oplus HFK_{\text{red}}(Y,K)\).