Splitting the spectral flow and the Alexander matrix
type: Item
instance of: scholarly article
title: Splitting the spectral flow and the Alexander matrix
zbMATH Open document ID: 0863.57008
DOI: 10.1007/BF02564494
author: Daniel Ruberman
author: Paul Kirk
author: E. Klassen
published in: Commentarii Mathematici Helvetici
publication date: 1997-06-10T00:00:00Z
full work available at URL: https://eudml.org/doc/140345
review text: This paper studies the problem of computing the spectral flow \(\text{SF} (\alpha, \beta)\) of the Atiyah-Patodi-Singer operator between two flat SU(2)-connexions \(\alpha, \beta\) on a 3-manifold \(Z\), when \(Z\) is split along a torus so that \(\alpha,\beta\) can be connected by a path of flat connexions on each piece. A well-known theorem of Yoshida shows that, if there are no singularities on the paths, \(\text{SF} (\alpha, \beta)\) only depends, in a simple explicit way, on the restriction of the paths to the splitting torus. This result has now been generalized by Cappell-Lee-Miller. In the present paper this problem is treated in cases where the pieces bound 4-manifolds (after filling in the torus with a solid torus) over which \(\alpha, \beta\) extend, by use of the Atiyah-Patodi-Singer Index theorem. In one application \(Z\) is a surgery on a satellite of a knot \(K\) and \(\alpha, \beta\) are reducible on the complement of a tubular neighborhood of \(K\). Then \(\alpha, \beta\) induce flat connexions on \(Z_0\), the manifold obtained by surgery along the same satellite of the trivial knot. The difference between the values of \(\text{SF} (\alpha, \beta)\) on \(Z\) and \(Z_0\) is computed and the result is a formula in terms of the (twisted) knot signatures of \(K\) defined by \(\alpha\) and \(\beta\) and the multiplicities of the corresponding roots of the Alexander polynomial. Since the restrictions to the torus are the same in both cases, this shows that Yoshida's formula requires a correction term when there are singularities. The formula of Cappell-Lee-Miller supplies such a correction in a more abstract context. Another application is to the case of graph manifolds where the two pieces of \(Z\) are Seifert fibred homology knot complements and uses methods of Fintushel-Stern.
Mathematics Subject Classification ID: 57M99
Mathematics Subject Classification ID: 58J20
zbMATH DE Number: 718588
zbMATH Keywords: spectral flow
zbMATH Keywords: Atiyah-Patodi-Singer operator
zbMATH Keywords: 3-manifold
zbMATH Keywords: 4-manifolds
zbMATH Keywords: surgery
zbMATH Keywords: satellite
zbMATH Keywords: knot
zbMATH Keywords: knot signatures
zbMATH Keywords: Alexander polynomial
zbMATH Keywords: Seifert fibred homology knot complements
reviewed by: Jerome Levine
MaRDI profile type: Publication
OpenAlex ID: W2086707717
description: scientific article; zbMATH DE number 718588