No abstract

No abstract

No 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

We define an extended Bloch group and show it is naturally isomorphic to H 3 (PSL(2, C) δ ; Z). Using the Rogers dilogarithm function this leads to an exact simplicial formula for the universal Cheeger-Chern-Simons class on this homology group. It also leads to an independent proof of the analytic relationship between volume and Chern-Simons invariant of hyperbolic 3-manifolds conjectured in [16] and proved in [24], as well as effective formulae for the Chern-Simons invariant of a hyperbolic 3-manifold.