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

Karim Adiprasito, Bruno Benedetti

Research output: Contribution to journalArticle

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 Sd; (2) there are exponentially many convex triangulations of the d-ball.

Original languageEnglish (US)
Pages (from-to)233-247
Number of pages15
JournalAnnales de l'Institut Henri Poincare (D) Combinatorics, Physics and their Interactions
Volume7
Issue number2
DOIs
StatePublished - 2020
Externally publishedYes

Keywords

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

ASJC Scopus subject areas

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

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