A Calculus for Plumbing Applied to the Topology of Complex Surface Singularities and Degenerating Complex Curves

Abstract

Abstract. Any graph-manifold can be obtained by plumbing according to some plumbing graph I\ A calculus for plumbing which includes normal forms for such graphs is developed. This is applied to answer several questions about the topology of normal complex surface singularities and analytic families of complex curves. For instance it is shown that the topology of the minimal resolution of a normal complex surface singularity is determined by the link of the singularity and even by its fundamental group if the singularity is not a cyclic quotient singularity or a cusp singularity.In this paper we describe a calculus for plumbed manifolds which lets one algorithmically determine when the oriented 3-manifolds M(TX) and A/fTj) obtained by plumbing according to two graphs Tx and T2 are homeomorphic (3-manifolds are oriented 3-manifolds throughout this paper, and homeomorphisms of 3-manifolds are orientation preserving). We then apply the calculus to answer several questions about the topology of isolated singularities of complex surfaces and one-parameter families of complex curves. These results are described below.Since the class of 3-manifolds obtainable by plumbing is precisely the class of graph-manifolds, which were classified, with minor exceptions, by Waldhausen [24], a calculus for plumbing is in some sense implicit in Waldhausen's work. Moreover, it has been known for some time that the calculus can be put in a form like the one given here, but the details have never appeared in the literature. A related calculus for plumbing trees has been worked by Bonahon and Siebenmann [1] in order to classify their "algebraic knots".We describe the calculus in greater generality than is needed for the present applications, since this involves minimal extra work, and the calculus is needed elsewhere ([4], [14], and [15]). In particular, in an appendix we describe two generalizations of it.The calculus consists of a collection of moves one can do to a plumbing graph T without altering the plumbed manifold M(T). To see these moves are sufficient, we describe how they can be used to reduce any graph to a normal form which is

Homological action of the modular group on some cubic moduli spaces

“…In fact (see [11]), the neighborhood boundary M 3 of any cyclic configuration of rational curves in a non-singular complex surface is a T 2 bundle over S 1 whose monodromy can be computed from the selfintersection numbers e i of the curves as…”

Combinatorics of rational singularities

“…The first problem is to give a proper definition of the complexity. Since rupture vertices of the dual intersection graph of the minimal good desingularization measure the local topological complexity of the link of a complex normal surface singularity (see [13]), it seems natural to define the complexity of a rational tree whose vertices have weights ≥ 2 to be the number of rupture vertices. This idea is enhanced by the following result: Remark 5.2.…”

“…Conversely, Neumann proved in [13] that the dual intersection graph associated with the minimal good desingularization of a normal surface singularity (algebraic or analytic) is determined by the topology of the surface in a neighbourhood of the singularity. So, to obtain a topological classification of rational singularities of complex surfaces, it is important to study the graphs which are the dual intersection graphs associated with a desingularization of these singularities.…”

Topological types and multiplicities of isolated quasi-homogeneous surface singularities

“…Hence in order to prove Theorem A, we may assume that both (V, 0) and (W, 0) are one of the seven classes above. Now we need two important results due to Neumann [12] and Orlik-Wagreich [13] about the abstract topology of these singularities. Neumann's results say that the minimal resolutions of these singularities are determined by the fundamental groups.…”