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34
The uniqueness of the spectral flow on spaces of unbounded self–adjoint Fredholm operators
, 2004
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Spectral invariants of operators of Dirac type on partitioned manifolds
 in Aspects of Boundary Problems in Analysis and Geometry, Editors
"... Abstract. We review the concepts of the index of a Fredholm operator, the spectral flow of a curve of selfadjoint Fredholm operators, the Maslov index of a curve of Lagrangian subspaces in symplectic Hilbert space, and the eta invariant of operators of Dirac type on closed manifolds and manifolds w ..."
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Cited by 12 (0 self)
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Abstract. We review the concepts of the index of a Fredholm operator, the spectral flow of a curve of selfadjoint Fredholm operators, the Maslov index of a curve of Lagrangian subspaces in symplectic Hilbert space, and the eta invariant of operators of Dirac type on closed manifolds and manifolds with boundary. We emphasize various (occasionally overlooked) aspects of rigorous definitions and explain the quite different stability properties. Moreover, we utilize the heat equation approach in various settings and show how these topological and spectral invariants are mutually related in the study of
Weak Symplectic Functional Analysis and General Spectral Flow Formula
, 2004
"... We consider a continuous curve of selfadjoint Fredholm extensions of a curve of closed symmetric operators with fixed minimal domain Dm and fixed intermediate domain DW. Our main example is a family of symmetric generalized operators of Dirac type on a compact manifold with boundary with varying w ..."
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Cited by 7 (3 self)
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We consider a continuous curve of selfadjoint Fredholm extensions of a curve of closed symmetric operators with fixed minimal domain Dm and fixed intermediate domain DW. Our main example is a family of symmetric generalized operators of Dirac type on a compact manifold with boundary with varying wellposed boundary conditions. Here DW is the first Sobolev space and Dm the subspace of sections with support in the interior. We express the spectral flow of the operator curve by the Maslov index of a corresponding curve of Fredholm pairs of Lagrangian subspaces of the quotient Hilbert space DW/Dm which is equipped with continuously varying weak symplectic structures induced by the Green form. In this paper, we specify the continuity conditions; define the Maslov index in weak symplectic analysis; discuss the required weak inner Unique Continuation Property; derive a General Spectral
Calderon projector for the Hessian of the perturbed ChernSimons function on a 3manifold with boundary
"... Abstract. The existence and continuity for the Calderón projector of the perturbed odd signature operator on a 3manifold is established. As an application we give a new proof of a result of Taubes relating the mod 2 spectral flow of a family of operators on a homology 3sphere with the difference i ..."
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Cited by 6 (4 self)
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Abstract. The existence and continuity for the Calderón projector of the perturbed odd signature operator on a 3manifold is established. As an application we give a new proof of a result of Taubes relating the mod 2 spectral flow of a family of operators on a homology 3sphere with the difference in local intersection numbers of the character varieties coming from a Heegard decomposition. 1.
Analytic Surgery of the ζdeterminant of the Dirac operator
 NUCLEAR PHYSICS B, TO APPEAR. OF THE ΖDETERMINANT AND SCATTERING THEORY
"... We review the work of the authors and their collaborators on the decomposition of the ζdeterminant of the Dirac operator into the contributions coming from different parts of a manifold. ..."
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Cited by 6 (2 self)
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We review the work of the authors and their collaborators on the decomposition of the ζdeterminant of the Dirac operator into the contributions coming from different parts of a manifold.
THE INTEGER VALUED SU(3) CASSON INVARIANT FOR BRIESKORN SPHERES
, 2003
"... We develop techniques for computing the integer valued SU(3) Casson invariant defined in [6]. Our method involves resolving the singularities in the flat moduli space using a twisting perturbation and analyzing its effect on the topology of the perturbed flat moduli space. These techniques, togeth ..."
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Cited by 5 (4 self)
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We develop techniques for computing the integer valued SU(3) Casson invariant defined in [6]. Our method involves resolving the singularities in the flat moduli space using a twisting perturbation and analyzing its effect on the topology of the perturbed flat moduli space. These techniques, together with BottMorse theory and the splitting principle for spectral flow, are applied to calculate τ SU(3)(Σ) for all Brieskorn homology spheres.
A general splitting formula for the spectral flow
 589–617 MR1721567 Geometry & Topology, Volume 9 (2005) Benjamin Himpel
, 1999
"... Abstract. We derive a decomposition formula for the spectral flow of a 1parameter family of selfadjoint Dirac operators on an odddimensional manifold M split along a hypersurface Σ (M = X ∪Σ Y). No transversality or stretching hypotheses are assumed and the boundary conditions can be chosen arbit ..."
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Abstract. We derive a decomposition formula for the spectral flow of a 1parameter family of selfadjoint Dirac operators on an odddimensional manifold M split along a hypersurface Σ (M = X ∪Σ Y). No transversality or stretching hypotheses are assumed and the boundary conditions can be chosen arbitrarily. The formula takes the form SF(D) = SF(DX, BX) + SF(DY, BY) + µ(BY, BX) + S where BX and BY are boundary conditions, µ denotes the Maslov index, and S is a sum of explicitly defined Maslov indices coming from stretching and rotating boundary conditions. The derivation is a simple consequence of Nicolaescu’s theorems and elementary properties of the Maslov index. We show how to use the formula and derive many of the splitting theorems in the literature as simple consequences. 1.
ON THE MASLOV INDEX OF LAGRANGIAN PATHS THAT ARE NOT TRANSVERSAL TO THE MASLOV CYCLE. SEMIRIEMANNIAN INDEX THEOREMS IN THE DEGENERATE CASE.
, 2003
"... ABSTRACT. The Maslov index of a Lagrangian path, under a certain transversality assumption, is given by an algebraic count of the intersections of the path with a subvariety of the Lagrangian Grassmannian called the Maslov cycle. In these notes we use the notion of generalized signatures at a singul ..."
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ABSTRACT. The Maslov index of a Lagrangian path, under a certain transversality assumption, is given by an algebraic count of the intersections of the path with a subvariety of the Lagrangian Grassmannian called the Maslov cycle. In these notes we use the notion of generalized signatures at a singularity of a smooth curve of symmetric bilinear forms to determine a formula for the computation of the Maslov index in the case of a realanalytic path having possibly non transversal intersections. Using this formula we give a general definition of Maslov index for continuous curves in the Lagrangian Grassmannian, both in the finite and in the infinite dimensional (Fredholm) case, and having arbitrary endpoints. Other notions of Maslov index are also considered, like the index for pairs of Lagrangian paths, the Kashiwara’s triple Maslov index, and Hörmander’s fourfold index. We discuss some applications of the theory, with special emphasis on the study of the Jacobi equation along a semiRiemannian geodesic. In this context, we prove an extension of several versions of the Morse index theorems for geodesics having possibly conjugate endpoints. CONTENTS
ON THE MASLOV INDEX OF SYMPLECTIC PATHS THAT ARE NOT TRANSVERSAL TO THE MASLOV CYCLE. SEMIRIEMANNIAN INDEX THEOREMS IN THE DEGENERATE CASE.
, 2003
"... ABSTRACT. The Maslov index of a symplectic path, under a certain transversality assumption, is given by an algebraic count of the intersections of the path with a subvariety of the Lagrangian Grassmannian called the Maslov cycle. In these notes we use the notion of generalized signatures at a singul ..."
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Cited by 4 (1 self)
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ABSTRACT. The Maslov index of a symplectic path, under a certain transversality assumption, is given by an algebraic count of the intersections of the path with a subvariety of the Lagrangian Grassmannian called the Maslov cycle. In these notes we use the notion of generalized signatures at a singularity of a smooth curve of symmetric bilinear forms to determine a formula for the computation of the Maslov index in the case of a realanalytic path having possibly non transversal intersections. Using this formula we give a general definition of Maslov index for continuous curves in the Lagrangian Grassmannian, both in the finite and in the infinite dimensional (Fredholm) case, and having arbitrary endpoints. Other notions of Maslov index are also considered, like the index for pairs of Lagrangian paths, the Kashiwara’s triple Maslov index, and the fourfold index. We discuss some applications of the theory, with special emphasis on the study of the Jacobi equation along a semiRiemannian geodesic. In this context, we prove an extension of several versions of the Morse index theorems for geodesics having possibly conjugate endpoints. CONTENTS