Geometrically bounding 3-manifold, volume and Betti number

TL;DR

This paper constructs specific hyperbolic 3-manifolds with prescribed Betti numbers and volume, which are uniquely bounding in the geometric sense, advancing understanding of hyperbolic 3-manifold boundaries.

Contribution

It introduces a method to construct hyperbolic 3-manifolds with given Betti numbers and volume that are geometrically bounding, using small cover theory and prior geometric results.

Findings

Constructed hyperbolic 3-manifolds with volume 16nv and Betti number k

Demonstrated existence of geometrically bounding hyperbolic 3-manifolds

Connected volume and Betti number properties through explicit constructions

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 $\beta^1(M)=k$ 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].