[Paper Review] Geometrically bounding 3-manifold, volume and Betti number

Jiming Ma, Fangting Zheng|arXiv (Cornell University)|Apr 10, 2017

Geometric and Algebraic Topology16 references4 citations

TL;DR

This paper constructs infinitely many closed hyperbolic 3-manifolds with prescribed odd first Betti numbers and volume $16nv$ (for $n \in \mathbb{Z}_+$) that geometrically bound totally geodesic hyperbolic 4-manifolds, using small cover theory over linearly glued dodecahedra and results from Kolpakov, Martelli, and Tschantz. The key contribution is proving that for each $n$, there are at least $\lceil(5n+3)/2\rceil$ distinct geometrically bounding 3-manifolds of volume $16nv$, showing that the number of such manifolds grows at least linearly with volume.

ABSTRACT

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 if it is the only boundary of a totally geodesic hyperbolic 4-manifold. However, there are very rare geometrically bounding closed hyperbolic 3-manifolds according to the previous research [11,13]. Let $v \approx 4.3062\ldots$ be the volume of the regular right-angled hyperbolic dodecahedron in $\mathbb{H}^{3}$, for each $n \in \mathbb{Z}_{+}$ and each odd integer $k$ in $[1,5n+3]$, we construct a closed hyperbolic 3-manifold $M$ with $β^1(M)=k$ and $vol(M)=16nv$ that bounds a totally geodesic hyperbolic 4-manifold. The proof uses small cover theory over a sequence of linearly-glued dodecahedra and some results of Kolpakov-Martelli-Tschantz [9].

Motivation & Objective

• To address the open question of which hyperbolic 3-manifolds geometrically bound totally geodesic hyperbolic 4-manifolds.
• To construct explicit families of closed hyperbolic 3-manifolds with prescribed first Betti number and volume that bound such 4-manifolds.
• To demonstrate that the number of geometrically bounding 3-manifolds per volume grows at least linearly, countering the rarity of such examples.

Proposed method

• Utilizes small cover theory over $n$-fold linearly glued dodecahedra to construct 3-manifolds with controlled topology.
• Employs $\mathbb{Z}_2^k$-colorings of dodecahedral facets to define small covers and compute Betti numbers via the $h$-vector of the orbit polytope.
• Applies results from Kolpakov, Martelli, and Tschantz on hyperbolic 4-manifolds with totally geodesic boundaries.
• Uses recursive relations on Betti numbers across sequences of glued dodecahedra to generate manifolds with odd Betti numbers in $[1, 5n+3]$.
• Constructs characteristic functions for $\mathbb{Z}_2^3$-colorings on a single dodecahedron to realize odd Betti numbers from 1 to 7.
• Combines the $\mathbb{Z}_2^3$-colorings with natural $\mathbb{Z}_2^4$-extensions to build the required 4-dimensional hyperbolic manifolds with totally geodesic boundary.

Experimental results

Research questions

• RQ1 For a hyperbolic 3-manifold with integer $\eta$-invariant, does it geometrically bound a totally geodesic hyperbolic 4-manifold?
• RQ2 How many geometrically bounding 3-manifolds exist with the same volume?
• RQ3 Can one systematically construct closed hyperbolic 3-manifolds with prescribed odd first Betti number and volume that bound hyperbolic 4-manifolds?

Key findings

• For each $n \in \mathbb{Z}_+$ and each odd integer $k \in [1, 5n+3]$, there exists a closed hyperbolic 3-manifold $M$ with $\beta^1(M) = k$ and $\mathrm{vol}(M) = 16nv$, where $v \approx 4.3062$ is the volume of the regular right-angled hyperbolic dodecahedron.
• The number of such geometrically bounding 3-manifolds of volume $16nv$ is at least $\lceil(5n+3)/2\rceil$, implying that the function $f_b(x)$, counting geometrically bounding 3-manifolds of volume $\leq x$, grows at least linearly.
• The construction relies on small covers over sequences of linearly glued dodecahedra, with $\mathbb{Z}_2^3$-colorings and their $\mathbb{Z}_2^4$-extensions.
• The Betti numbers of the resulting 3-manifolds are systematically controlled via recursive relations on the number of glued dodecahedra and coloring parameters.
• Explicit $\mathbb{Z}_2^3$-colorings on a single dodecahedron are provided that realize all odd Betti numbers from 1 to 7.
• The 3-manifolds constructed are proven to bound totally geodesic hyperbolic 4-manifolds, confirming their geometric bounding property.