A Hyperbolic 4-Manifold

Davis, Michael W.

doi:10.2307/2044771

Cited by 30 publications

(39 citation statements)

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“…To our knowledge, these are the first known examples of such hyperbolic manifolds in higher dimensions. Our primary example is the Davis hyperbolic 4-manifold [9], which was shown to have a spin structure in [18]. Our method is to use the G-spin theorem (the G-index theorem for the Dirac operator [6]) where G is a group of orientation preserving isometries of a Riemannian spin manifold M .…”

Section: Introductionmentioning

confidence: 99%

Harmonic spinors on the Davis hyperbolic 4-manifold

2019

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In this paper we use the G-spin theorem to show that the Davis hyperbolic 4-manifold admits harmonic spinors. This is the first example of a closed hyperbolic 4-manifold that admits harmonic spinors. We also explicitly describe the Spinor bundle of a spin hyperbolic 2-or 4-manifold and show how to calculated the subtle sign terms in the G-spin theorem for an isometry, with isolated fixed points, of a closed spin hyperbolic 2-or 4-manifold.1991 Mathematics Subject Classification. 20H10, 22E40, 30F40, 53C27, 57M50, 58J05, 58J20.

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Section: Introductionmentioning

confidence: 99%

Harmonic spinors on the Davis hyperbolic 4-manifold

2019

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“…The 120-cell is a notable convex regular 4-polytope that admits an embedding in H 4 as a right-angled polytope. Identifying opposite 3-dimensional faces of the 120-cell gives a compact hyperbolic 4-manifold [Dav85]. The dual of the 120-cell is the 600-cell whose boundary is a flag complex that is built of 600 tetrahedra and is characterized by the property that the link of each vertex is an icosahedron.…”

Section: B Some High Density Bipartite Examplesmentioning

confidence: 99%

Virtually Fibering Right-Angled Coxeter Groups

Jankiewicz¹,

Norin²,

Wise³

2019

J. Inst. Math. Jussieu

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We show that certain right-angled Coxeter groups have finite index subgroups that quotient to Z with finitely generated kernels. The proof uses Bestvina-Brady Morse theory facilitated by combinatorial arguments. We describe a variety of examples where the plan succeeds or fails. Among the successful examples are the right-angled reflection groups in H 4 with fundamental domain the 120-cell or the 24-cell.

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“…The role of the abelian surface in our situation is played by a beautiful hyperbolic 4manifold called the Davis manifold [8] (see also the description in [26]). This manifold is constructed by gluing opposite 3-faces of a 120-cell in H 4 , a certain regular fourdimensional polyhedron (so-called because it has 120 three-dimensional faces, each a regular dodecahedron).…”

Section: H 4 and The Small Resolutionmentioning

confidence: 99%

Hyperbolic geometry and non-Kähler manifolds with trivial canonical bundle

Fine

Panov

2010

Geom. Topol.

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We use hyperbolic geometry to construct simply connected symplectic or complex manifolds with trivial canonical bundle and with no compatible Kähler structure. We start with the desingularisations of the quadric cone in C 4 : the smoothing is a natural S 3 -bundle over H 3 , its holomorphic geometry is determined by the hyperbolic metric; the small-resolution is a natural S 2 -bundle over H 4 with symplectic geometry determined by the metric. Using hyperbolic geometry, we find orbifold quotients with trivial canonical bundle; smooth examples are produced via crepant resolutions.In particular, we find the first example of a simply connected symplectic 6-manifold with c 1 D 0 that does not admit a compatible Kähler structure. We also find infinitely many distinct complex structures on 2.S 3 S 3 / # .S 2 S 4 / with trivial canonical bundle. Finally, we explain how an analogous construction for hyperbolic manifolds in higher dimensions gives symplectic non-Kähler "Fano" manifolds of dimension 12 and higher.53D35, 32Q55; 51M10, 57M25

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A Hyperbolic 4-Manifold

Cited by 30 publications

References 0 publications

Harmonic spinors on the Davis hyperbolic 4-manifold

Harmonic spinors on the Davis hyperbolic 4-manifold

Virtually Fibering Right-Angled Coxeter Groups

Hyperbolic geometry and non-Kähler manifolds with trivial canonical bundle

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