Self-adjoint elliptic operators and manifold decompositions part II: Spectral flow and maslov index

Sylvain E. Cappell, Ronnie Lee, Edward Y. Miller

Research output: Contribution to journalArticle

Abstract

This the second part of a three-part investigation of the behavior of certain analytical invariants of manifolds that can be split into the union of two submanifolds. In Part I we studied a splicing construction for low eigenvalues of self-adjoint elliptic operators over such a manifold. Here we go on to study parameter families of such operators and use the previous "static" results in obtaining results on the decomposition of spectral flows. Some of these "dynamic" results are expressed in terms of Maslov indices of Lagrangians. The present treatment is sufficiently general to encompass the difficulties of zero-modes at the ends of the parameter families as well as that of "jumping Lagrangians." In Part III, we will compare infinite- and finite-dimensional Lagrangians and determinant line bundles and then introduce "canonical perturbations" of Lagrangian subvarieties of symplectic varieties. We shall then use this information to study invariants of 3-manifolds, including Casson's invariant.

Original languageEnglish (US)
Pages (from-to)869-909
Number of pages41
JournalCommunications on Pure and Applied Mathematics
Volume49
Issue number9
StatePublished - Sep 1996

Fingerprint

Maslov Index
Spectral Flow
Information use
Self-adjoint Operator
Elliptic Operator
Decomposition
Decompose
Casson Invariant
Invariant
Line Bundle
Submanifolds
Determinant
Union
Eigenvalue
Perturbation
Zero
Operator
Family

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Self-adjoint elliptic operators and manifold decompositions part II : Spectral flow and maslov index. / Cappell, Sylvain E.; Lee, Ronnie; Miller, Edward Y.

In: Communications on Pure and Applied Mathematics, Vol. 49, No. 9, 09.1996, p. 869-909.

Research output: Contribution to journalArticle

@article{3759896ad8744861b8d17b27bf16a676,
title = "Self-adjoint elliptic operators and manifold decompositions part II: Spectral flow and maslov index",
abstract = "This the second part of a three-part investigation of the behavior of certain analytical invariants of manifolds that can be split into the union of two submanifolds. In Part I we studied a splicing construction for low eigenvalues of self-adjoint elliptic operators over such a manifold. Here we go on to study parameter families of such operators and use the previous {"}static{"} results in obtaining results on the decomposition of spectral flows. Some of these {"}dynamic{"} results are expressed in terms of Maslov indices of Lagrangians. The present treatment is sufficiently general to encompass the difficulties of zero-modes at the ends of the parameter families as well as that of {"}jumping Lagrangians.{"} In Part III, we will compare infinite- and finite-dimensional Lagrangians and determinant line bundles and then introduce {"}canonical perturbations{"} of Lagrangian subvarieties of symplectic varieties. We shall then use this information to study invariants of 3-manifolds, including Casson's invariant.",
author = "Cappell, {Sylvain E.} and Ronnie Lee and Miller, {Edward Y.}",
year = "1996",
month = "9",
language = "English (US)",
volume = "49",
pages = "869--909",
journal = "Communications on Pure and Applied Mathematics",
issn = "0010-3640",
publisher = "Wiley-Liss Inc.",
number = "9",

}

TY - JOUR

T1 - Self-adjoint elliptic operators and manifold decompositions part II

T2 - Spectral flow and maslov index

AU - Cappell, Sylvain E.

AU - Lee, Ronnie

AU - Miller, Edward Y.

PY - 1996/9

Y1 - 1996/9

N2 - This the second part of a three-part investigation of the behavior of certain analytical invariants of manifolds that can be split into the union of two submanifolds. In Part I we studied a splicing construction for low eigenvalues of self-adjoint elliptic operators over such a manifold. Here we go on to study parameter families of such operators and use the previous "static" results in obtaining results on the decomposition of spectral flows. Some of these "dynamic" results are expressed in terms of Maslov indices of Lagrangians. The present treatment is sufficiently general to encompass the difficulties of zero-modes at the ends of the parameter families as well as that of "jumping Lagrangians." In Part III, we will compare infinite- and finite-dimensional Lagrangians and determinant line bundles and then introduce "canonical perturbations" of Lagrangian subvarieties of symplectic varieties. We shall then use this information to study invariants of 3-manifolds, including Casson's invariant.

AB - This the second part of a three-part investigation of the behavior of certain analytical invariants of manifolds that can be split into the union of two submanifolds. In Part I we studied a splicing construction for low eigenvalues of self-adjoint elliptic operators over such a manifold. Here we go on to study parameter families of such operators and use the previous "static" results in obtaining results on the decomposition of spectral flows. Some of these "dynamic" results are expressed in terms of Maslov indices of Lagrangians. The present treatment is sufficiently general to encompass the difficulties of zero-modes at the ends of the parameter families as well as that of "jumping Lagrangians." In Part III, we will compare infinite- and finite-dimensional Lagrangians and determinant line bundles and then introduce "canonical perturbations" of Lagrangian subvarieties of symplectic varieties. We shall then use this information to study invariants of 3-manifolds, including Casson's invariant.

UR - http://www.scopus.com/inward/record.url?scp=0008993194&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0008993194&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0008993194

VL - 49

SP - 869

EP - 909

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 9

ER -