Gauge theory for embedded surfaces. I
type: Item
instance of: scholarly article
title: Gauge theory for embedded surfaces. I
zbMATH Open document ID: 0799.57007
DOI: 10.1016/0040-9383(93)90051-V
published in: Topology
publication date: 1994-05-12T00:00:00Z
review text: Every 2-dimensional homology class of a closed oriented 4-manifold is represented by an embedded surface. In this paper, the authors give lower bounds on the genus of such a surface in terms of its self-intersection number. They work in a smooth simply-connected 4-manifold which has non- trivial polynomial invariants with \(b_ 2^ +\) odd and not less than 3. As an application, the ``local Thom conjecture'' is solved. It states that if \(C\) is a smooth algebraic curve in complex 2-space and \(B^ 4\) is an embedded 4-ball whose boundary meets \(C\) transversely in a knot \(K\) then the surface \(C\cap B^ 4\) realizes the slice genus of \(K\). This result implies that the unknotting number of a knot which arises from singularities in algebraic curves equals the genus of the Milnor fibre.
reviewed by: Peter Teichner
Mathematics Subject Classification ID: 57N13
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57R57
Mathematics Subject Classification ID: 57R95
zbMATH DE Number: 550344
zbMATH Keywords: gauge theory
zbMATH Keywords: algebraic knots
zbMATH Keywords: local Thom conjecture
zbMATH Keywords: 2-dimensional homology class
zbMATH Keywords: closed oriented 4-manifold
zbMATH Keywords: embedded surface
zbMATH Keywords: polynomial invariants
zbMATH Keywords: knot
zbMATH Keywords: slice genus
zbMATH Keywords: unknotting number
zbMATH Keywords: genus of the Milnor fibre
author: P. B. Kronheimer
author: T. S. Mrowka
MaRDI profile type: Publication
Recommended article: Gauge theory for embedded surfaces. II
Recommended article: Q4952548
Recommended article: Gauge theory on compact surfaces
Recommended article: Surface operators in gauge theory
Recommended article: Gauge theories, tessellations \& Riemann surfaces
Recommended article: Pure gauge theory of the Cosserat surface
Recommended article: Q4888003
Recommended article: Quantum gauge theory on compact surfaces
Recommended article: Gauge theory, isotropy, and surfaces in affine 4-space
Recommended article: Q3975785
description: scientific article; zbMATH DE number 550344