\(\underline{SW}\)=Milnor torsion
type: Item
instance of: scholarly article
title: \(\underline{SW}\)=Milnor torsion
zbMATH Open document ID: 0870.57018
author: Guowu Meng
author: Clifford Henry Taubes
published in: Mathematical Research Letters
publication date: 1997-07-29T00:00:00Z
reviewed by: Jerome Levine
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 980880
zbMATH Keywords: Seiberg-Witten invariant
zbMATH Keywords: 3-manifolds
zbMATH Keywords: Milnor torsion
zbMATH Keywords: homology orientation
zbMATH Keywords: \(\text{spin}^{\mathbb{C}}\)-structures
zbMATH Keywords: link
zbMATH Keywords: Alexander polynomial
zbMATH Keywords: skein theory
zbMATH Keywords: Euler structure
MaRDI profile type: Publication
DOI: 10.4310/MRL.1996.V3.N5.A8
review text: This paper announces the result that a version of the Seiberg-Witten invariant for certain smooth \(3\)-manifolds coincides with the Milnor torsion, as refined by Turaev. Let \(X\) be a compact oriented \(3\)-manifold such that \(b_1 (=\dim H^1 (X))>0\) and \(\partial X\) is a (possibly empty) union of tori, equipped with a homology orientation (if \(\partial X=\emptyset\) there is a canonical homology orientation given by Poincaré duality). Let \(\mathcal S\) denote the set of \(\operatorname{spin}^{\mathbb C}\)-structures on \(X\) such that \(\det s|\partial X\) is trivial. Consider pairs \((s,x)\) where \(s\in\mathcal S\) and \(x\in H=H^2 (X,\partial X )/\)torsion is a lift of the Chern class \(c_1 (\det s)\in H^2 (X)\). The Seiberg-Witten equations determined by \(s\) will have solutions \((A,\psi )\) where \(A\) is a connexion on \(\det s\). The curvature \(F_A\) determines an element of \(H\) which is a lift of \(c_1 (\det s)\). In the generic case the solution is an oriented \(0\)-dimensional manifold and \(SW(s,x)\) is defined to be the algebraic sum of all the solutions for which \(F_A =x\). If \(b_1 =1\) and \(\partial X=\emptyset\), then it is necessary to also choose a generator of \(H\). Now \(H^2 (X,\partial X)\) acts transitively on \(\mathcal S\) and so, if one chooses a `base' element \(y=(s_0 ,x_0 )\), one can define NEWLINE\[CARRIAGE_RETURNNEWLINESW_y (X)=\sum_{g\in H}\left(\sum_{h\equiv g}SW(h\cdot s_0,x_0 +2g)\right)g\in \mathbb ZH CARRIAGE_RETURNNEWLINE\]NEWLINE where \(h\) ranges over the (finite) set of lifts of \(g\) into \(H^2 (X,\partial X)\) and \(\mathbb Z H\) is the set of all \(\mathbb Z\)-valued functions on \(H\). Then, to remove the dependency on \(y\), define \(\underline{SW}(X)\) to be the orbit of \(SW_y (X)\) in \(\mathbb Z H/H\). In fact if \(b_1 >1\), then \(SW_y (X)\in\mathbb Z [H]\), where \(\mathbb Z [H]\) is the ring of finitely-supported \(\mathbb Z \)-valued functions on \(H\), i.e. the integral group ring of…
description: scientific article; zbMATH DE number 980880