Computing spectral flow via cup products
type: Item
instance of: scholarly article
title: Computing spectral flow via cup products
zbMATH Open document ID: 0816.58038
published in: Journal of Differential Geometry
publication date: 1995-02-26T00:00:00Z
review text: The authors investigate the spectral flow of the ``Dirac'' operator \(D_ A = *d_ A - d_ A*\) acting on \(0 + 2\) forms on a 3-manifold \(M\) as \(A\) varies in the connections in an SU(2)-bundle over \(M\). The main result states that if \(A_ t\) is a path of flat connections such that \(\text{Ker }D_{A_ t} = 0\) for \(t = 0\), then the spectral flow at time \(t = 0\) (i.e. the slopes of the eigenvalues of \(D_{A_ t}\) that are crossing 0 at \(t = 0\)) can be expressed in terms of the eigenvalues of a bilinear form which is a composition of two cup products in the cohomology of \(M\) with twisted coefficients. In particular, the authors compute the spectral flow \(\bmod 4\) between two representations of the fundamental group of a 3-manifold which fibers over the circle with fiber a torus.
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 727340
zbMATH Keywords: Dirac operator
zbMATH Keywords: spectral flow
zbMATH Keywords: 3-manifold
author: Paul Kirk
author: E. Klassen
reviewed by: Georgi E. Karadzhov
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214455777
OpenAlex ID: W1596208710
Wikidata QID: Q115179018
DOI: 10.4310/JDG/1214455777
description: scientific article; zbMATH DE number 727340