### 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 language | English (US) |
---|---|

Pages (from-to) | 869-909 |

Number of pages | 41 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 49 |

Issue number | 9 |

State | Published - Sep 1996 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*,

*49*(9), 869-909.

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

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 49, no. 9, pp. 869-909.

}

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 -