Geometrically bounding 3–manifolds, volume and Betti numbers

Ma, Jiming; Zheng, Fangting

doi:10.2140/agt.2023.23.1055

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Geometrically bounding $3$–manifolds, volume and Betti numbers

Jiming Ma and Fangting Zheng

Algebraic & Geometric Topology 23 (2023) 1055–1096

DOI: 10.2140/agt.2023.23.1055

Abstract

A hyperbolic $3$ –manifold is geometrically bounding if it is the only boundary of a totally geodesic hyperbolic $4$ –manifold. According to previous results of Long and Reid (2000) and Meyerhoff and Neumann (1992), geometrically bounding closed hyperbolic $3$ –manifolds are very rare. Assume the value $v \approx 4.3062 \dots$ for the volume of the regular right-angled hyperbolic dodecahedron $P$ in $ℍ^{3}$ . For each positive integer $n$ and each odd integer $k$ in $[1, 5 n + 3]$ , we construct a closed hyperbolic $3$ –manifold $M$ with $β^{1} (M) = k$ and $vol (M) = 16 n v$ which bounds a totally geodesic hyperbolic $4$ –manifold. In particular, for every positive odd integer $k$ , there are infinitely many geometrically bounding $3$ –manifolds whose first Betti numbers are $k$ . The proof exploits the real toric manifold theory over a sequence of stacking dodecahedra, together with some results obtained by Kolpakov, Martelli and Tschantz (2015).

Keywords

hyperbolic 3-manifolds, geometrically bounding, hyperbolic 4-manifolds, small cover

Mathematical Subject Classification

Primary: 57R90, 57M50, 57S25

Supplementary material

Figures with supplemental calculations

References

Publication

Received: 9 March 2020

Revised: 16 November 2020

Accepted: 30 December 2020

Published: 6 June 2023

Authors

Jiming Ma
School of Mathematical Sciences Fudan University Shanghai China

Fangting Zheng
Department of Pure Mathematics Xi’an Jiaotong-Liverpool University Suzhou China

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Volume 23, issue 3 (2023)