On the gluing problem for the \(\eta\)-invariant
type: Item
instance of: scholarly article
title: On the gluing problem for the \(\eta\)-invariant
zbMATH Open document ID: 0821.58037
author: Ulrich Bunke
published in: Journal of Differential Geometry
publication date: 1995-09-19T00:00:00Z
review text: Consider a generalized Dirac operator \(D\) over a compact Riemannian manifold \(M\) that is partitioned by a compact hypersurface \(N\) such that \(M := M_ 1 \cup_ N M_ 2\). Assume that the Riemannian metric of \(M\) and \(D\) have a product structure near \(N\), i.e. \(D = I(\partial/\partial r + D_ N)\) with some Dirac operator \(D_ N\) on \(N\). Using boundary conditions of Atiyah-Patodi-Singer-type parametrized by Lagrangian subspaces \(L_ i\) of \(\text{ker }D_ N\) one defines selfadjoint extensions \(D_ i\), \(i = 1,2\) over \(M_ i\). The main result is a formula expressing the \(\eta\)-invariant of \(D\) as a sum of the \(\eta\)- invariants of \(D_ i\), an invariant \(m(L_ 1,L_ 2)\) of the pair of the Lagrangian subspaces \(L_ 1\), \(L_ 2\) which is related to the Maslov index and an integer valued term \(J\). In the adiabatic limit, i.e. if a tubular neighbourhood of \(N\) is long enough, the vanishing of \(J\) is shown under certain regularity conditions. This result is applied in order to prove cutting and pasting formulas for the \(\eta\)-invariant, a Wall non-additivity result for the index of Atiyah-Patodi-Singer boundary value problems and a splitting formula for the spectral flow.
Mathematics Subject Classification ID: 58J20
Mathematics Subject Classification ID: 58J60
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 58J32
zbMATH DE Number: 761169
zbMATH Keywords: gluing problem
zbMATH Keywords: eta invariant
zbMATH Keywords: global boundary conditions
zbMATH Keywords: spectral flow
zbMATH Keywords: Dirac operator
Wikidata QID: Q115178596
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214456222
OpenAlex ID: W1578520505
DOI: 10.4310/JDG/1214456222
description: scientific article; zbMATH DE number 761169