Spectral flow and intersection number
type: Item
instance of: scholarly article
title: Spectral flow and intersection number
zbMATH Open document ID: 0784.46052
author: Nobukazu Otsuki
author: Kenro Furutani
published in: Journal of Mathematics of Kyoto University
publication date: 1994-03-22T00:00:00Z
review text: Let \(F= F(\mathbb{H})\) be the set of bounded Fredholm operators on a separable Hilbert space \(\mathbb{H}\) of infinite dimension. A subset \(\mathfrak F\) of \(F\) consisting of selfadjoint operators has three components: \({\mathfrak F}= {\mathfrak F}_ + +{\mathfrak F}_ - +{\mathfrak F}_*\) (\({\mathfrak F}_ +({\mathfrak F}_ -)\) consists of essentially positive (negative) operators and \({\mathfrak F}_*\) consists of others). \({\mathfrak F}_*\) is the classifying space for the \(K^{-1}\)-group, especially there are the spectral flows: \(\pi_ 1({\mathfrak F}_*)\to\mathbb{Z}\). The Bott periodicity theorem for complex \(K\)-group says that the spectral flows and the Fredholm index are equivalent as topological invariants through the suspension or the desuspension. However, there are analytic difficulties in dealing with the spectral flows arising from the family of differential operators with different domains of definitions. In this paper the authors construct an embedding of the unitary group \(\gamma: U(N)\to{\mathfrak F}_*\) for each \(N\) satisfying some conditions and give a non-trivial example of the spectral flow arising from a variation of non-local elliptic boundary conditions imposed on a fixed elliptic differential operator and its spectral flow formula in terms of intersection numbers between certain singular cycles.
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 46M20
zbMATH DE Number: 408746
zbMATH Keywords: index theory
zbMATH Keywords: selfadjoint operator
zbMATH Keywords: \(K\)-theory
zbMATH Keywords: bounded Fredholm operators
zbMATH Keywords: Bott periodicity
zbMATH Keywords: complex \(K\)-group
zbMATH Keywords: non-local elliptic boundary conditions
zbMATH Keywords: elliptic differential operator
zbMATH Keywords: spectral flow formula
zbMATH Keywords: singular cycles
reviewed by: V. M. Deundjak
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250519347
OpenAlex ID: W1915626314
DOI: 10.1215/KJM/1250519347
description: scientific article; zbMATH DE number 408746