TY - JOUR

T1 - Geometrically bounding 3–manifolds, volume and Betti numbers

AU - Ma, Jiming

AU - Zheng, Fangting

N1 - Funding Information:
Jiming Ma was partially supported by NSFC 11771088 and 12171092. Fangting Zheng was supported by NSFC 12101504 and XJTLU Research Development Fund RDF-19-01-29. The authors appreciate greatly the referees and Alastair Darby for their valuable and constructive comments on improving the text, which made our paper much more precise and readable.
Publisher Copyright:
© 2023 MSP (Mathematical Sciences Publishers).

PY - 2023

Y1 - 2023

N2 - 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 H3. 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).

AB - 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 H3. 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).

UR - http://www.scopus.com/inward/record.url?scp=85163214093&partnerID=8YFLogxK

U2 - 10.2140/agt.2023.23.1055

DO - 10.2140/agt.2023.23.1055

M3 - Article

AN - SCOPUS:85163214093

SN - 1472-2747

VL - 23

SP - 1055

EP - 1096

JO - Algebraic and Geometric Topology

JF - Algebraic and Geometric Topology

IS - 3

ER -