A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves.

*(English)*Zbl 0546.57002In this important paper the author describes and gives numerous applications of the algorithm which allows one to decide whether two 3-manifolds obtained by plumbing according to a graph of \(S^ 1\)-bundles over compact surfaces (possibly with boundary) are homeomorphic. Namely the manifolds associated with two graphs are the same if and only if there is a sequence of moves (which belong to 8 types) which transforms one graph into the other. Practical applications are based on the use of normal forms to which any graph can be reduced and such that manifolds corresponding to different graphs are distinct. As the author points out, this calculus is implicit in F. Waldhausen’s work [Invent. Math. 3, 308-333; ibid. 4, 87-117 (1967; Zbl 0168.445)] on classification of graph manifolds. The author earlier used this calculus to define an integral invariant of plumbed homology spheres [Lect. Notes Math. 788, 125-144 (1980; Zbl 0436.57002)]. Closely related to the author’s calculus is Bonahon and Siebenmann’s census of oriented diffeomorphism types of manifolds arising from weighted trees. This technique is applied to the analysis of two types of 3-manifolds naturally appearing in algebraic geometry. The first type is the singularity links, i.e., the boundaries of regular neighbourhoods of isolated singularities of complex surfaces.

The main result is that with two exceptions the fundamental group of the singularity link determines the genera, normal bundles and intersection numbers of a minimal good resolution of the singularity (i.e., the resolution in which all intersections are normal crossings, no more than two curves intersect in one point and there are no \(CP^ 1\)’s with self-intersection \(-1\)). The exceptional singularity links which do not determine the resolution are the lens spaces \(L(p,q)\) and the torus bundles over circles with monodromy \(A\in SL_ 2({\mathbb{Z}})\) such that trace \(A\) is \(\geq 3\). Among other facts the author shows that the singularity links are irreducible 3-manifolds (Problem 3.20 in R. Kirby’s list [Proc. Symp. Pure Math. 32, Part 2, 273-312 (1978; Zbl 0394.57002)]) and that the resolution of a singularity is star-shaped provided the singularity link is a Seifert manifold. Another interesting result is that a complex surface V is topologically the suspension of a closed 3-manifold if and only if it is homeomorphic to an Inoue surface [M. Inoue, Complex Anal. algebr. Geom. 91-106 (1977; Zbl 0365.14011)].

The second type of 3-manifold arising from algebraic geometry is the link of families of curves. The latter are the manifolds of the form \(\pi^{- 1}(\partial D)\), where \(\pi: W\to D\) is an analytic map of a complex surface on the unit disk such that all fibres except the one over the origin are nonsingular complete algebraic curves. The main result is that the fundamental group of the link of a minimal good family of curves defines the numerical type of the family, i.e., genera, normal bundles and intersection numbers of components.

The main result is that with two exceptions the fundamental group of the singularity link determines the genera, normal bundles and intersection numbers of a minimal good resolution of the singularity (i.e., the resolution in which all intersections are normal crossings, no more than two curves intersect in one point and there are no \(CP^ 1\)’s with self-intersection \(-1\)). The exceptional singularity links which do not determine the resolution are the lens spaces \(L(p,q)\) and the torus bundles over circles with monodromy \(A\in SL_ 2({\mathbb{Z}})\) such that trace \(A\) is \(\geq 3\). Among other facts the author shows that the singularity links are irreducible 3-manifolds (Problem 3.20 in R. Kirby’s list [Proc. Symp. Pure Math. 32, Part 2, 273-312 (1978; Zbl 0394.57002)]) and that the resolution of a singularity is star-shaped provided the singularity link is a Seifert manifold. Another interesting result is that a complex surface V is topologically the suspension of a closed 3-manifold if and only if it is homeomorphic to an Inoue surface [M. Inoue, Complex Anal. algebr. Geom. 91-106 (1977; Zbl 0365.14011)].

The second type of 3-manifold arising from algebraic geometry is the link of families of curves. The latter are the manifolds of the form \(\pi^{- 1}(\partial D)\), where \(\pi: W\to D\) is an analytic map of a complex surface on the unit disk such that all fibres except the one over the origin are nonsingular complete algebraic curves. The main result is that the fundamental group of the link of a minimal good family of curves defines the numerical type of the family, i.e., genera, normal bundles and intersection numbers of components.

##### MSC:

57N10 | Topology of general \(3\)-manifolds (MSC2010) |

32S05 | Local complex singularities |

14E15 | Global theory and resolution of singularities (algebro-geometric aspects) |

57M05 | Fundamental group, presentations, free differential calculus |

57R19 | Algebraic topology on manifolds and differential topology |

14J30 | \(3\)-folds |

14J25 | Special surfaces |

32J25 | Transcendental methods of algebraic geometry (complex-analytic aspects) |

14H15 | Families, moduli of curves (analytic) |

14B05 | Singularities in algebraic geometry |

14E20 | Coverings in algebraic geometry |

##### Keywords:

plumbing of circle bundles over compact surfaces; 3-manifolds; classification of graph manifolds; boundaries of regular neighbourhoods of isolated singularities of complex surfaces; fundamental group of the singularity link; lens spaces; torus bundles over circles; Seifert manifold; Inoue surface; link of families of curves
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\textit{W. D. Neumann}, Trans. Am. Math. Soc. 268, 299--343 (1981; Zbl 0546.57002)

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##### References:

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