A Hodge-Type Theorem for Manifolds with Fibered Cusp Metrics

Müller, Jörn

doi:10.1007/s00039-011-0115-x

Cited by 5 publications

(7 citation statements)

References 23 publications

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“…However, we are interested in understanding how to study manifolds also that generalise symmetric spaces. This question of how better to characterise boundary fibre bundles with this property was studied by J. Mueller in [10], and he is taking it up again together with the authors of this paper in current work.…”

Section: The "Split" Property For the Gauss-bonnet Operator And The H...mentioning

confidence: 94%

“…One of the main differences between Vaillant's work and the work in this paper is that certain geometric obstructions arise in the construction of the parametrix of the (second order) Hodge Laplacian that do not arise in the construction of the parametrix of the (first order) Dirac operator. These obstructions have arisen in work of J. Mueller [10] that studies the Hodge Laplacian on φ-manifolds from a perturbation theory viewpoint. Thus it is interesting to see how the obstructions also arise using a pseudodifferential approach.…”

Section: Introductionmentioning

confidence: 99%

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A Parametrix Construction for the Laplacian on ℚ-rank 1 Locally Symmetric Spaces

Grieser

Hunsicker

2013

Fourier Analysis

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This paper presents the construction of parametrices for the Gauss-Bonnet and Hodge Laplace operators on noncompact manifolds modelled on Q-rank 1 locally symmetric spaces. These operators are, up to a scalar factor, φ-differential operators, that is, they live in the generalised φ-calculus studied by the authors in a previous paper, which extends work of Melrose and Mazzeo. However, because they are not totally elliptic elements in this calculus, it is not possible to construct parametrices for these operators within the φ-calculus. We construct parametrices for them in this paper using a combination of the b-pseudodifferential operator calculus of R. Melrose and the φ-pseudodifferential operator calculus. The construction simplifies and generalizes the construction done by Vaillant in his thesis for the Dirac operator. In addition, we study the mapping properties of these operators and determine the appropriate Hlibert spaces between which the Gauss-Bonnet and Hodge Laplace operators are Fredholm. Finally, we establish regularity results for elements of the kernels of these operators. (2000). Primary 58AJ40; Secondary 35J75. Mathematics Subject Classification

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Section: The "Split" Property For the Gauss-bonnet Operator And The H...mentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

A Parametrix Construction for the Laplacian on ℚ-rank 1 Locally Symmetric Spaces

Grieser

Hunsicker

2013

Fourier Analysis

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“…Our Theorem 5.1 is closely related to results of [Dai91] and [Mül11]. Dai and Müller work with Riemannian E, B and a Riemannian submersion π : E → B.…”

Section: Introductionmentioning

confidence: 60%

Isometric group actions and the cohomology of flat fiber bundles

Banagl¹

2013

Groups Geom. Dyn.

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1, which corresponds to local angular forms on the sphere bundle, and which survives to E n , to the Euler class of the sphere bundle. Since SO(n) is compact, the spectral sequence of the bundle collapses at E 2 , by the theorem. Thus d n = 0 and we obtain the following corollary:Corollary. The real Euler class of an oriented, flat, linear sphere bundle (structure group SO(n)) over a closed, smooth manifold is zero.We thus obtain a topological proof, without using the Chern-Weil theory of curvature forms, of a result closely related to the results of [Mil58, Section 4], which do rely on Chern-Weil theory.By [Smi77], there is a flat manifold M 2n with nonzero Euler characteristic for every n > 1. If the associated tangent sphere bundle of M, with structure group SO(2n), were flat, then the Euler class of such a sphere bundle would vanish according to the above corollary. We arrive at:

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“…Just a few such papers are [25], [23], [22], [1], [9], [10], [11]. The most relevant to this paper are [17], in which the author of the present paper and her collaborators studied L 2 harmonic forms on fibred cusp manifolds and their relationship to middle perversity intersection cohomology groups, and [27], in which harmonic forms on fibred cusp manifolds satisfying the same geometric restriction as in this paper are found which represent relative and absolute cohomology groups. At the same time, many authors have been involved with the development of new analytic tools for the study of singular and noncompact spaces.…”

Section: Introductionmentioning

confidence: 99%

Extended Hodge theory for fibred cusp manifolds

Hunsicker

2018

J. Topol. Anal.

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For a particular class of pseudo manifolds, we show that the intersection cohomology groups for any perversity may be naturally represented by extended weighted L 2 harmonic forms for a complete metric on the regular stratum with respect to some weight determined by the perversity. Extended weighted L 2 harmonic forms are harmonic forms that are almost in the given weighted L 2 space for the metric in question, but not quite. This result is akin to the representation of absolute and relative cohomology groups for a manifold with boundary by extended harmonic forms on the associated manifold with cylindrical ends. In analogy with that setting, in the unweighted L 2 case, the boundary values of the extended harmonic forms define a Lagrangian splitting of the boundary space in the long exact sequence relating upper and lower middle perversity intersection cohomology groups.

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A Hodge-Type Theorem for Manifolds with Fibered Cusp Metrics

Cited by 5 publications

References 23 publications

A Parametrix Construction for the Laplacian on ℚ-rank 1 Locally Symmetric Spaces

A Parametrix Construction for the Laplacian on ℚ-rank 1 Locally Symmetric Spaces

Isometric group actions and the cohomology of flat fiber bundles

Extended Hodge theory for fibred cusp manifolds

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