A cheeger-type exponential bound for the number of triangulated manifolds

Karim Adiprasito; Bruno Benedetti

doi:10.4171/AIHPD/85

A cheeger-type exponential bound for the number of triangulated manifolds

Karim Adiprasito, Bruno Benedetti

Einstein Institute of Mathematics

The Hebrew University of Jerusalem

Research output: Contribution to journal › Article › peer-review

Abstract

In terms of the number of triangles, it is known that there are more than exponentially many triangulations of surfaces, but only exponentially many triangulations of surfaces with bounded genus. In this paper we provide a first geometric extension of this result to higher dimensions. We show that in terms of the number of facets, there are only exponentially many geometric triangulations of space forms with bounded geometry in the sense of Cheeger (curvature and volume bounded below, and diameter bounded above). This establishes a combinatorial version of Cheeger’s finiteness theorem. Further consequences of our work are: (1) there are exponentially many geometric triangulations of S^d; (2) there are exponentially many convex triangulations of the d-ball.

Original language	English
Pages (from-to)	233-247
Number of pages	15
Journal	Annales de l'Institut Henri Poincare (D) Combinatorics, Physics and their Interactions
Volume	7
Issue number	2
DOIs	https://doi.org/10.4171/AIHPD/85
State	Published - 2020

Keywords

Bounded geometry
Collapsibility
Discrete finiteness Cheeger theorem
Discrete quantum gravity
Geometric manifolds
Simple homotopy theory
Triangulations

All Science Journal Classification (ASJC) codes

Statistical and Nonlinear Physics
Algebra and Number Theory
Statistics and Probability
Geometry and Topology
Discrete Mathematics and Combinatorics

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10.4171/AIHPD/85

Cite this

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title = "A cheeger-type exponential bound for the number of triangulated manifolds",

abstract = "In terms of the number of triangles, it is known that there are more than exponentially many triangulations of surfaces, but only exponentially many triangulations of surfaces with bounded genus. In this paper we provide a first geometric extension of this result to higher dimensions. We show that in terms of the number of facets, there are only exponentially many geometric triangulations of space forms with bounded geometry in the sense of Cheeger (curvature and volume bounded below, and diameter bounded above). This establishes a combinatorial version of Cheeger{\textquoteright}s finiteness theorem. Further consequences of our work are: (1) there are exponentially many geometric triangulations of Sd; (2) there are exponentially many convex triangulations of the d-ball.",

keywords = "Bounded geometry, Collapsibility, Discrete finiteness Cheeger theorem, Discrete quantum gravity, Geometric manifolds, Simple homotopy theory, Triangulations",

author = "Karim Adiprasito and Bruno Benedetti",

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doi = "10.4171/AIHPD/85",

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AU - Adiprasito, Karim

AU - Benedetti, Bruno

N1 - Publisher Copyright: © European Mathematical Society.

PY - 2020

Y1 - 2020

N2 - In terms of the number of triangles, it is known that there are more than exponentially many triangulations of surfaces, but only exponentially many triangulations of surfaces with bounded genus. In this paper we provide a first geometric extension of this result to higher dimensions. We show that in terms of the number of facets, there are only exponentially many geometric triangulations of space forms with bounded geometry in the sense of Cheeger (curvature and volume bounded below, and diameter bounded above). This establishes a combinatorial version of Cheeger’s finiteness theorem. Further consequences of our work are: (1) there are exponentially many geometric triangulations of Sd; (2) there are exponentially many convex triangulations of the d-ball.

AB - In terms of the number of triangles, it is known that there are more than exponentially many triangulations of surfaces, but only exponentially many triangulations of surfaces with bounded genus. In this paper we provide a first geometric extension of this result to higher dimensions. We show that in terms of the number of facets, there are only exponentially many geometric triangulations of space forms with bounded geometry in the sense of Cheeger (curvature and volume bounded below, and diameter bounded above). This establishes a combinatorial version of Cheeger’s finiteness theorem. Further consequences of our work are: (1) there are exponentially many geometric triangulations of Sd; (2) there are exponentially many convex triangulations of the d-ball.

KW - Bounded geometry

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A cheeger-type exponential bound for the number of triangulated manifolds

Abstract

Keywords

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