Download this article
 Download this article For screen
For printing
Recent Issues

Volume 24
Issue 2, 595–1223
Issue 1, 1–594

Volume 23, 9 issues

Volume 22, 8 issues

Volume 21, 7 issues

Volume 20, 7 issues

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editorial Interests
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Author Index
To Appear
 
Other MSP Journals
Geometrically bounding $3$–manifolds, volume and Betti numbers

Jiming Ma and Fangting Zheng

Algebraic & Geometric Topology 23 (2023) 1055–1096
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 4.3062 for the volume of the regular right-angled hyperbolic dodecahedron P in 3. For each positive integer n and each odd integer k in [1,5n + 3], we construct a closed hyperbolic 3–manifold M with β1(M) = k and  vol(M) = 16nv 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

Open Access made possible by participating institutions via Subscribe to Open.