arXiv research

Four hyperbolic 24-cell 4-manifolds with one cusp are identified.

Estimates intersection pairing in hyperbolic 4-manifolds.

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 subtl…

Built the smallest non-commensurable hyperbolic 4-manifold.

Plumbing of surfaces embeds in hyperbolic 4-manifolds.

We construct here two new examples of non-orientable, non-compact, hyperbolic 4-manifolds. The first has minimal volume $v_m = 4π^2/3$ and two cusps. This example has the lowest number of cusps among known minimal volume hyperbolic 4-manifolds. The second has volume $2\cdot v_m$ and one cusp. It has lowest volume among…

4 flat 3-manifolds realized in hyperbolic 4-space.

This note shows every integer can be a signature of a hyperbolic 4-manifold.

The paper explores cusp types in hyperbolic 4-manifolds and their commensurability classes.

We prove that there are at least 2 commensurability classes of minimal-volume hyperbolic 4-manifolds. Moreover, by applying a well-known technique due to Gromov and Piatetski-Shapiro, we build the smallest known non-arithmetic hyperbolic 4-manifold.

Paper computes G-index for specific hyperbolic manifolds.

We provide, for hyperbolic and flat 3-manifolds, obstructions to bounding hyperbolic 4-manifolds, thus resolving in the negative a question of Farrell and Zdravkovska.

No spin structures found in a hyperbolic 4D space.

New proof shows 4-manifolds can't support complex structures.

We prove that for any \e>0, there exists a closed hyperbolic 4-manifold with a closed geodesic of length < \e.

The study finds minimal volume hyperbolic 4-manifolds with embedded 3-manifolds.

We develop a way of seeing a complete orientable hyperbolic $4$-manifold $\mathcal{M}$ as an orbifold cover of a Coxeter polytope $\mathcal{P} \subset \mathbb{H}^4$ that has a facet colouring. We also develop a way of finding totally geodesic sub-manifolds $\mathcal{N}$ in $\mathcal{M}$, and describing the result of mu…

New hyperbolic 4-manifolds found with special functions.

We construct examples of codimension two hyperbolic link complements in closed smooth 4-manifolds with homeomorphism type $\#_{2k}S^2 \times S^2$. All our examples are based on a construction of J. Ratcliffe and S. Tschantz, who constructed 1171 non-compact finite volume hyperbolic 4-manifolds of minimal volume. We the…

Study counts hyperbolic 4-manifolds with vanishing Seiberg-Witten invariants up to volume $v$.

We show how to construct a Kirby diagram for a large class of finite volume hyperbolic 4-manifolds constructed by J. Ratcliffe and S. Tschantz.

4-manifolds show every flat 3-manifold as cusp sections.

A 4-manifold is constructed with some curious metric properties; or maybe it is many 4-manifolds masquerading as one, which would explain why it looks curious. Anyway, knots in the 3-sphere with complete finite volume hyperbolic metrics on their complements play a role in this story.

We show the existence of hyperbolic 4-manifolds with vanishing Seiberg-Witten invariants, addressing a conjecture of Claude LeBrun. This is achieved by showing, using results in geometric and arithmetic group theory, that certain hyperbolic 4-manifolds contain L-spaces as hypersurfaces.

It is known that the volume function for hyperbolic manifolds of dimension $\geq 3$ is finite-to-one. We show that the number of nonhomeomorphic hyperbolic 4-manifolds with the same volume can be made arbitrarily large. This is done by constructing a sequence of finite-sided finite-volume polyhedra with side-pairings t…

Proves properties of 4-manifolds with scalar curvature constraints.

First example of a hyperbolic 4-orbifold underlying $\mathbb{P}^2$.

The paper constructs a hyperbolic 4-manifold with rational homology sphere cusp sections.

We show that some hyperbolic 3-manifolds which are tessellated by copies of the regular ideal hyperbolic tetrahedron embed geodesically in a complete, finite volume, hyperbolic 4-manifold. This allows us to prove that the complement of the figure-eight knot geometrically bounds a complete, finite volume hyperbolic 4-ma…

Many noncompact hyperbolic 3-manifolds are topologically complements of links in the 3-sphere. Generalizing to dimension 4, we construct a dozen examples of noncompact hyperbolic 4-manifolds, all of which are topologically complements of varying numbers of tori and Klein bottles in the 4-sphere. Finite covers of some o…

It is observed that on many 4-manifolds there is a unique smooth structure underlying a globally hyperbolic Lorentz metric. For instance, every contractible smooth 4-manifold admitting a globally hyperbolic Lorentz metric is diffeomorphic to the standard $\R^4$. Similarly, a smooth 4-manifold homeomorphic to the produc…

New 4-manifolds found without certain Einstein metrics, using Seiberg-Witten theory.

In this paper, for each finite group $G$, we construct explicitly a non-compact complete finite-volume arithmetic hyperbolic $4$-manifold $M$ such that $\mathrm{Isom}\,M \cong G$, or $\mathrm{Isom}^{+}\,M \cong G$. In order to do so, we use essentially the geometry of Coxeter polytopes in the hyperbolic $4$-space, on o…

Since there is no hyperbolic Dehn filling theorem for higher dimensions, it is challenging to construct explicit hyperbolic manifolds of small volume in dimension at least four. Here, we build up closed hyperbolic 4-manifolds of volume $\frac{34π^2}{3}\cdot 16$ by using the small cover theory. In particular, we classif…

The paper constructs Poincaré-Einstein 4-manifolds with various cusps.

We prove that every complete finite-volume hyperbolic 3-manifold $M$ that is tessellated into (embedded) right-angled regular polyhedra (dodecahedra or ideal octahedra) embeds geodesically in a complete finite-volume connected orientable hyperbolic 4-manifold $W$, which is also tessellated into right-angled regular pol…

6 out of 29 dodecahedral 3-spheres are L-spaces, solving a Seiberg-Witten question.

Paper shows Seiberg-Witten invariants vanish for Davis hyperbolic 4-manifold.

It is well known that an arbitrary closed orientable $3$-manifold can be realized as the unique boundary of a compact orientable $4$-manifold, that is, any closed orientable $3$-manifold is cobordant to zero. In this paper, we consider the geometric cobordism problem: a hyperbolic $3$-manifold is geometrically bounding…

We exhibit the first examples of compact orientable hyperbolic manifolds that do not have any spin structure. We show that such manifolds exist in all dimensions $n \geq 4$. The core of the argument is the construction of a compact orientable hyperbolic $4$-manifold $M$ that contains a surface $S$ of genus $3$ with sel…

This note computes the "renormalized volume" and a renormalizedGauss-Bonnet-Chern formula for the Euler characteristic ofasymptotically complex hyperbolic Einstein (in short: ACHE)4-manifolds.

The study proves properties of 4D projective manifolds and builds non-hyperbolic examples.

The study constructs geometrically decomposable aspherical 4-manifolds with non-zero signature and explores their properties.

We obtain an ordering of closed aspherical 4-manifolds that carry a non-hyperbolic Thurston geometry. As application, we derive that the Kodaira dimension of geometric 4-manifolds is monotone with respect to the existence of maps of non-zero degree.

Paper finds new 3D shapes that can be inside a 4D space.

New estimates are derived concerning the behavior of self-dual hamonic 2-forms on a compact Riemannian 4-manifold with non-trivial Seiberg-Witten invariants. Applications include a vanishing theorem for certain Seiberg-Witten invariants on compact 4-manifolds of constant negative sectional curvature.

A finite-volume hyperbolic 3-manifold geometrically bounds if it is the geodesic boundary of a finite-volume hyperbolic 4-manifold. We construct here an example of non-compact, finite-volume hyperbolic 3-manifold that geometrically bounds. The 3-manifold is the complement of a link with eight components, and its volume…

Using techniques from the theory of Kirby calculus we give an explicit construction of a four dimensional hyperbolic link complement in a 4-manifold that is diffeomorphic to the standard 4-sphere.