On the number of finite \(p/q\)-surgeries (Q2792144)

This paper shows that for a given order of the first homology group, only finitely many \(3\)-manifolds with non-cyclic, finite fundamental group are obtained by Dehn surgery on knots in the \(3\)-sphere \(S^3\).NEWLINENEWLINEIt has been proved by \textit{J. Morgan} and \textit{G. Tian} [Ricci flow and the Poincaré conjecture. Providence, RI: American Mathematical Society (AMS); Cambridge, MA: Clay Mathematics Institute (2007; Zbl 1179.57045)] that every \(3\)-manifold with finite fundamental group is a spherical manifold. These have been classified into five families by \textit{H. Seifert} [Acta Math. 60, 147--238 (1933; Zbl 0006.08304)]: lens spaces, dihedral manifolds (i.e., prism manifolds), tetrahedral manifolds, octahedral manifolds and icosahedral manifolds. This implies that a \(3\)-manifold with non-cyclic, finite fundamental group is one of the last four manifolds. Moreover, all the last three manifolds are obtained by Dehn surgery on torus knots due to \textit{L. Moser}'s result [Pac. J. Math. 38, 737--745 (1971; Zbl 0202.54701)]. We know already that for a fixed order of the first homology, only finitely many such \(3\)-manifolds are obtained by Dehn surgery on knots in the \(3\)-sphere. Hence this paper studies dihedral manifolds, say \(Y_n\), with the Seifert invariant \((-1;1/2,1/2,m/n)\) such that \(n>2m>0\).NEWLINENEWLINEThe author first gives a recursive relationship among the Heegaard Floer \(d\)-invariants for the manifolds \(Y_n\). As a corollary, for a given natural number \(N\) there are only finitely many manifolds \(Y_n\) with \(|d(Y_n,\sigma)|\leq N\) for all \(\sigma\in \mathrm{Spin}^{\mathrm{c}}(Y_n)\). For a knot \(K\subset S^3\) let \(S^3_{p/q}(K)\) be the \(3\)-manifold obtained by \(p/q\)-surgery on \(K\). Since dihedral manifolds are \(L\)-spaces, the author next shows that for a given positive integer \(m\) and for any knot \(K\) such that \(S^3_{4m}(K)\) is an \(L\)-space, \(|d(S^3_{p/q}(K),\sigma)|\leq 4m\) for all \(\sigma\in \mathrm{Spin}^{\mathrm{c}}(S^3_{4m}(K))\).