Topology of Hypersurface Singularities
نویسنده
چکیده
Kähler’s paper Über die Verzweigung einer algebraischen Funktion zweier Veränderlichen in der Umgebung einer singulären Stelle” offered a more perceptual view of the link of a complex plane curve singularity than that provided shortly before by Brauner. Kähler’s innovation of using a “square sphere” became standard in the toolkit of later researchers on singularities. We describe his contribution and survey developments since then, including a brief discussion of the topology of isolated hypersurface singularities in higher dimension. 1. Topology of plane curve singularities The Riemann surface of an algebraic function on the plane represents a complex curve (real dimension 2) as a covering of the Riemann sphere, ramified over some finite collection of points. At the start of the 20th century, the study of complex surfaces (real dimension 4) was rapidly developing, and they too were often studied as “Riemann surfaces,” — now of algebraic functions on the complex plane. The branching of such a “Riemann surface” is along a complex curve, and the only difficult case in understanding the local topology of this branching is at a singularity of the curve. The problem therefore arose, to understand the topology of a complex plane curve C near a singular point. The first discussion of this appears to be in Heegaard’s 1898 thesis [16] (see Epple [13, 14]). A small ball B around the singular point will intersect the curve C in a set that is homeomorphic to the cone on C ∩ ∂B. This set C ∩ ∂B, which is a link (disjoint union of embedded circles) in the 3-sphere ∂B, therefore determines the local topology completely. It thus suffices to understand the links that arise this way: links of plane curve singularities, as they are now called. To understand the local branching of the “Riemann surface” one also needs the fundamental group of the complement of the link in the 3-sphere. The first comprehensive article on this topic is the 1928 paper [3] by Karl Brauner, who writes that he learned the problem from Wirtinger, who had spoken on it to the Mathematikervereinigung in Meran in 1905 and subsequently held a seminar on the topic in Vienna. In his paper Brauner follows Heegaard in using stereographic projection to move the link from S to R. He then describes the topology of the link in terms of repeated cabling, and gives an explicit presentation of the fundamental group of the complement of the link.
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تاریخ انتشار 2014