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## The Eta Invariant And Families Of Pseudodifferential Operators (1995)

Venue: | MR 96h:58169 |

Citations: | 41 - 7 self |

### Citations

620 | Spectral asymmetry in Riemannian geometry, I, II and III,
- Atiyah, Patodi, et al.
- 1975
(Show Context)
Citation Context ...ach component of the elliptic set. Introduction The eta invariant of the spin Dirac operator, and of the signature operator, on an odd dimensional manifold was introduced by Atiyah, Patodi and Singer =-=[1-=-] as the boundary correction term for their index formula on an even-dimensional compact manifold with boundary. Their denition extends directly to all `admissible' Dirac operators and was later shown... |

311 | The Atiyah-Patodi-Singer index theorem
- Melrose
- 1993
(Show Context)
Citation Context ... operators on compact manifolds without boundary. In the Dirac setting there are various further extensions to non-compact manifolds (by Bruning and Seeley [6], by Muller [16], by Stern [19] and in [1=-=3-=-]), to singular manifolds (by Cheeger [8]) to boundary problems (by Branson and Gilkey [5], by Douglas and Wojciechowski [9], by Lesch and Wojciechowski [11] and by Muller [17]), to families (by Bismu... |

133 |
Spectral asymmetry and Riemannian
- Atiyah, Patodi, et al.
- 1975
(Show Context)
Citation Context ...ach component of the elliptic set. Introduction The eta invariant of the spin Dirac operator, and of the signature operator, on an odd dimensional manifold was introduced by Atiyah, Patodi and Singer =-=[1]-=-as the boundary correction term for their index formula on an even-dimensional compact manifold with boundary. Their definition extends directly to all ‘admissible’ Dirac operators and was later shown... |

79 |
The analysis of elliptic families
- Bismut, Freed
- 1986
(Show Context)
Citation Context ...he connection dening the Dirac operator is Cliord and unitary. The crucial property of such an admissible Dirac operator is embodied in the local index theorem, see the argument of Bismut and Freed [4=-=]-=- or the discussion in [13]. As for the Dirac case, under variation of metric or connection within the class of admissible Dirac operators, the variation of the eta functional is local. More precisely ... |

79 | Families of Dirac operators, boundaries and the b-calculus
- Melrose, Piazza
- 1997
(Show Context)
Citation Context ...heeger [8]) to boundary problems (by Branson and Gilkey [5], by Douglas and Wojciechowski [9], by Lesch and Wojciechowski [11] and by Muller [17]), to families (by Bismut and Cheeger [3], [2] and in [=-=-=-14], [15] ) also to dene `higher' eta invariants (by Lott [12], by Getzler [10] and by Wu [20]). Here a somewhat dierent `pseudodierential' extension of the eta invariant is given. This is closely rel... |

43 |
Eta invariants and their adiabatic limits
- Bismut, Cheeger
- 1989
(Show Context)
Citation Context ...s paper is Theorem 1. For any compact manifold Y and any vector bundle E over Y , there is an additive homomorphism η from the ring of invertible elliptic elements Inv∗ ⊂ Ψ ∗ sus(Y ; E) into C, i.e., =-=(2)-=- η :Inv∗ −→ C, η(A ◦ B) =η(A)+η(B), and if ð is an admissible Dirac operator acting on the sections of some Clifford module E over Y then provided ð is invertible η(iDt + ð) is the eta invariant of ð ... |

34 |
An index theorem for first order regular singular operators
- Brüning, Seeley
- 1988
(Show Context)
Citation Context ...all self-adjoint elliptic pseudodifferential operators on compact manifolds without boundary. In the Dirac setting there are various further extensions to non-compact manifolds (by Brüning and Seeley =-=[6]-=-, by Müller [16], by Stern [19] and in [13]), to singular manifolds (by Cheeger [8]) to boundary problems (by Branson and Gilkey [5], by Douglas and Wojciechowski [9], by Lesch and Wojciechowski [11] ... |

34 |
Adiabatic limits of the η–invariants. The odd– dimensional Atiyah–Patodi–Singer problem
- Douglas, Wojciechowski
- 1991
(Show Context)
Citation Context ... manifolds (by Brüning and Seeley [6], by Müller [16], by Stern [19] and in [13]), to singular manifolds (by Cheeger [8]) to boundary problems (by Branson and Gilkey [5], by Douglas and Wojciechowski =-=[9]-=-, by Lesch and Wojciechowski [11] and by Müller [17]), to families (by Bismut and Cheeger [3], [2] and in [14], [15] ) also to define ‘higher’ eta invariants (by Lott [12], by Getzler [10] and by Wu [... |

30 | Residues of the eta function for an operator of Dirac type
- Branson, Gilkey
- 1992
(Show Context)
Citation Context ...urther extensions to non-compact manifolds (by Bruning and Seeley [6], by Muller [16], by Stern [19] and in [13]), to singular manifolds (by Cheeger [8]) to boundary problems (by Branson and Gilkey [5], by Douglas and Wojciechowski [9], by Lesch and Wojciechowski [11] and by Muller [17]), to families (by Bismut and Cheeger [3], [2] and in [14], [15] ) also to dene `higher' eta invariants (by Lott... |

24 |
η-invariants, the adiabatic approximation and conical singularities. I. The adiabatic approximation
- Cheeger
- 1987
(Show Context)
Citation Context ...oundary. In the Dirac setting there are various further extensions to non-compact manifolds (by Bruning and Seeley [6], by Muller [16], by Stern [19] and in [13]), to singular manifolds (by Cheeger [8=-=-=-]) to boundary problems (by Branson and Gilkey [5], by Douglas and Wojciechowski [9], by Lesch and Wojciechowski [11] and by Muller [17]), to families (by Bismut and Cheeger [3], [2] and in [14], [15]... |

23 |
Private communication
- Mazzeo
(Show Context)
Citation Context ...ls of an operator. It is a suspended analogue of the `residue trace' of Guillemin and Wodzicki. Similar trace functionals have also been considered by Burns and Mazzeo in work on Hochschild homology, =-=[-=-7]. Theorem 2. If A s 2sm sus (Y ; E) is a smooth family of elliptic and invertible elements then d ds (A s ) = 2 i f Tr dA s ds A 1 s : (4) The fact that f Tr(A) is local, i.e. symbolic, has the imp... |

17 |
Higher Eta-Invariants, K-Theory 6
- Lott
- 1992
(Show Context)
Citation Context ... by Douglas and Wojciechowski [9], by Lesch and Wojciechowski [11] and by Muller [17]), to families (by Bismut and Cheeger [3], [2] and in [14], [15] ) also to dene `higher' eta invariants (by Lott [1=-=-=-2], by Getzler [10] and by Wu [20]). Here a somewhat dierent `pseudodierential' extension of the eta invariant is given. This is closely related to Singer's comments in [18] on the formal analogy betw... |

14 | Cyclic homology and the Atiyah-Patodi-Singer index theorem
- Getzler
- 1993
(Show Context)
Citation Context ...ojciechowski [9], by Lesch and Wojciechowski [11] and by Muller [17]), to families (by Bismut and Cheeger [3], [2] and in [14], [15] ) also to dene `higher' eta invariants (by Lott [12], by Getzler [1=-=-=-0] and by Wu [20]). Here a somewhat dierent `pseudodierential' extension of the eta invariant is given. This is closely related to Singer's comments in [18] on the formal analogy between the index fun... |

12 |
On the η-invariant of generalized Atiyah–Patodi–Singer boundary value problems
- Lesch, Wojciechowski
- 1996
(Show Context)
Citation Context ...y [6], by Müller [16], by Stern [19] and in [13]), to singular manifolds (by Cheeger [8]) to boundary problems (by Branson and Gilkey [5], by Douglas and Wojciechowski [9], by Lesch and Wojciechowski =-=[11]-=- and by Müller [17]), to families (by Bismut and Cheeger [3], [2] and in [14], [15] ) also to define ‘higher’ eta invariants (by Lott [12], by Getzler [10] and by Wu [20]). Here a somewhat different ‘... |

8 |
Manifolds with cusps of rank one, spectral theory and L2-index theorem
- Müller
- 1987
(Show Context)
Citation Context ... elliptic pseudodierential operators on compact manifolds without boundary. In the Dirac setting there are various further extensions to non-compact manifolds (by Bruning and Seeley [6], by Muller [16=-=-=-], by Stern [19] and in [13]), to singular manifolds (by Cheeger [8]) to boundary problems (by Branson and Gilkey [5], by Douglas and Wojciechowski [9], by Lesch and Wojciechowski [11] and by Muller [... |

8 |
L 2 -index theorems on locally symmetric spaces
- Stern
(Show Context)
Citation Context ...odierential operators on compact manifolds without boundary. In the Dirac setting there are various further extensions to non-compact manifolds (by Bruning and Seeley [6], by Muller [16], by Stern [19=-=-=-] and in [13]), to singular manifolds (by Cheeger [8]) to boundary problems (by Branson and Gilkey [5], by Douglas and Wojciechowski [9], by Lesch and Wojciechowski [11] and by Muller [17]), to famili... |

7 |
The η-invariant and the index, in: Mathematical Aspects of String Theory
- Singer
- 1988
(Show Context)
Citation Context ...r' eta invariants (by Lott [12], by Getzler [10] and by Wu [20]). Here a somewhat dierent `pseudodierential' extension of the eta invariant is given. This is closely related to Singer's comments in [1=-=-=-8] on the formal analogy between the index function and the eta invariant (despite their obvious dierences). For purposes of comparison considersrst the familiar properties of the index. Let (Y ) be ... |

4 |
On the eta-invariant of generalized Atiyah-Patodi-Singer boundary value problems
- Lesch, Wojciechowski
- 1996
(Show Context)
Citation Context ... [6], by Muller [16], by Stern [19] and in [13]), to singular manifolds (by Cheeger [8]) to boundary problems (by Branson and Gilkey [5], by Douglas and Wojciechowski [9], by Lesch and Wojciechowski [=-=-=-11] and by Muller [17]), to families (by Bismut and Cheeger [3], [2] and in [14], [15] ) also to dene `higher' eta invariants (by Lott [12], by Getzler [10] and by Wu [20]). Here a somewhat dierent `p... |

3 |
An index theorem for order regular singular operators
- Bruning, Seeley
- 1988
(Show Context)
Citation Context ...all self-adjoint elliptic pseudodierential operators on compact manifolds without boundary. In the Dirac setting there are various further extensions to non-compact manifolds (by Bruning and Seeley [6=-=-=-], by Muller [16], by Stern [19] and in [13]), to singular manifolds (by Cheeger [8]) to boundary problems (by Branson and Gilkey [5], by Douglas and Wojciechowski [9], by Lesch and Wojciechowski [11]... |

3 |
K.P.: Adiabatic limits of the ^-invariants. The odd-dimensional Atiyah-Patodi-Singer problem
- Douglas, Wojciechowski
- 1989
(Show Context)
Citation Context ...anifolds (by Bruning and Seeley [6], by Muller [16], by Stern [19] and in [13]), to singular manifolds (by Cheeger [8]) to boundary problems (by Branson and Gilkey [5], by Douglas and Wojciechowski [9], by Lesch and Wojciechowski [11] and by Muller [17]), to families (by Bismut and Cheeger [3], [2] and in [14], [15] ) also to dene `higher' eta invariants (by Lott [12], by Getzler [10] and by Wu [... |

3 |
invariants and Manifolds with boundary
- Eta
- 1994
(Show Context)
Citation Context ...], by Stern [19] and in [13]), to singular manifolds (by Cheeger [8]) to boundary problems (by Branson and Gilkey [5], by Douglas and Wojciechowski [9], by Lesch and Wojciechowski [11] and by Muller [=-=-=-17]), to families (by Bismut and Cheeger [3], [2] and in [14], [15] ) also to dene `higher' eta invariants (by Lott [12], by Getzler [10] and by Wu [20]). Here a somewhat dierent `pseudodierential' ex... |

1 |
The Chern character and the -invariant of the Dirac operator, KTheory
- Wu
- 1993
(Show Context)
Citation Context ...], by Lesch and Wojciechowski [11] and by Muller [17]), to families (by Bismut and Cheeger [3], [2] and in [14], [15] ) also to dene `higher' eta invariants (by Lott [12], by Getzler [10] and by Wu [2=-=-=-0]). Here a somewhat dierent `pseudodierential' extension of the eta invariant is given. This is closely related to Singer's comments in [18] on the formal analogy between the index function and the e... |

1 |
The Chern character and the η-invariant of the Dirac operator, K-Theory
- Wu
- 1993
(Show Context)
Citation Context ...], by Lesch and Wojciechowski [11] and by Müller [17]), to families (by Bismut and Cheeger [3], [2] and in [14], [15] ) also to define ‘higher’ eta invariants (by Lott [12], by Getzler [10] and by Wu =-=[20]-=-). Here a somewhat different ‘pseudodifferential’ extension of the eta invariant is given. This is closely related to Singer’s comments in [18]on the formal analogy between the index function and the ... |