The Topology of Surface Singularities

Abstract

We consider a reduced complex surface germ (X, p). We do not assume that X is normal at p, and so, the singular locus (Σ, p) of (X, p) could be one dimensional. This text is devoted to the description of the topology of (X, p). By the conic structure theorem (see [19]), (X, p) is homeomorphic to the cone on its link LX. First of all, for any good resolution ρ : (Y, EY) → (X, 0) of (X, p), there exists a factorization through the normalization ν : (X,p) → (X, 0) (see [12] Thm. 3.14). This is why we proceed in two steps. 1) When (X, p) a normal germ of surface, p is an isolated singular point and the link LX of (X, p) is a well defined differentiable three-manifold. Using the good minimal resolution of (X, p), LX is given as the boundary of a well defined plumbing (see Section 2) which has a negative definite intersection form (see [11] and [21]). 2) In Section 3, we use a suitably general morphism, π : (X, p) → (C 2 , 0), to describe the topology of a surface germ (X, p) which has a 1-dimensional singular locus (Σ, p). We give a detailed description of the quotient morphism induced by the normalization ν on the link LX of (X,p) (see also Section 2 in Luengo-Pichon [15]). In Section 4, we give a detailed proof of the existence of a good resolution of a normal surface germ by the Hirzebruch-Jung method (Theorem 4.2.1). With this method a good resolution is obtained via an embedded resolution of the discriminant of π (see [10]). An example is given Section 6. An appendix (Section 5) is devoted to the topological study of lens spaces and to the description of the minimal resolution of quasi-ordinary singularities of surfaces. Section 5 provides the necessary background material to make the proof of Theorem 4.2.1 self-contained.

The topology of the normalization of complex surface germs

“…In Section 2, we describe the topology of a d-curling and the topology of a singular pinched torus which is defined as the mapping torus of an orientation preserving homeomorphism acting on a reducible germ of curves. Curlings and singular pinched tori are already studied in [5]. But, to prove the new results 3.0.3, 3.0.2 and 4.0.1 of this paper, we need to insist on particular properties, given in 2.0.3, of these two topological objects.…”

“…In Section 3 of [5], one can find a description of the topology of N (K Σ+ ) which implies the following lemma 3.0.1. To be be self-contained, we begin Section 3 with a quick proof of it.…”

“…By Lemma 3.0.1 (or Proposition 3.12 in [5]), if L X is a topological manifold the normalization ν : (X ′ , p ′ ) → (X, p) is a homeomorphism. Then the link L X ′ is also simply connected and by Mumford's theorem [7] (X ′ , p ′ ) is smooth at p ′ .…”

“…In Section 3, we show that each connected component of N (K Σ+ ) is a singular pinched torus (see also [5]). As stated in [4] and as proved in [5], It implies that ν restricted to ν −1 (L X \ K Σ+ ) is the composition of two kind of quotients: curlings and identifications.…”

“…In Section 3, we show that each connected component of N (K Σ+ ) is a singular pinched torus (see also [5]). As stated in [4] and as proved in [5], It implies that ν restricted to ν −1 (L X \ K Σ+ ) is the composition of two kind of quotients: curlings and identifications. Here, we need the more detailed description of N (K Σ+ ) given Section 2, to prove the following Lemma 3.0.2 which shows how the topology of L X determines the invariants n(σ) and d j , 1 ≤ j ≤ n(σ), of the normalization morphism.…”