Cobordism of algebraic knots
type: Item
instance of: scholarly article
title: Cobordism of algebraic knots
zbMATH Open document ID: 0669.57020
DOI: 10.1007/BF01393975
author: Susan Szczepanski
published in: Inventiones Mathematicae
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143677
review text: Any complex isolated hypersurface singularity f: (C\({}^{n+1},0)\to (C,0)\) determines an algebraic knot in \(S^{2n+1}\), the link of the singularity. The author shows that the cobordism type of an algebraic knot depends on the Dynkin diagram with respect to only weakly distinguished bases. More precisely: If there exist weakly distinguished bases for the middle homology of the Milnor fibers of two functions with isolated singularities at the origin such that the Dynkin diagrams summarizing the intersection forms coincide, then the singularities of the functions are cobordant. In particular there exist topological distinct algebraic knots which are cobordant. In order to prove the main result, the author introduces the ``link of a perturbation'' and, as a first step, constructs and analyzes a cobordism between the complement of these links of two singularities with common diagram. Then this cobordism is replaced by another one (which is obtained by surgery modifications) such that the new cobordism relating the link complements is an (integer) homology product. This cobordism is extended to an h-cobordism from \(S^{2n+1}\) to itself which is diffeomorphic by the h-cobordism theorem to \(S^{2n+1}\times I\). In particular the knots are cobordant.
Mathematics Subject Classification ID: 57R90
Mathematics Subject Classification ID: 32S05
Mathematics Subject Classification ID: 57Q45
zbMATH DE Number: 4096400
zbMATH Keywords: complex isolated hypersurface singularity
zbMATH Keywords: algebraic knot
zbMATH Keywords: cobordism type of an algebraic knot
zbMATH Keywords: Dynkin diagram
zbMATH Keywords: bases for the middle homology of the Milnor fibers
reviewed by: András Némethi
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4096400