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Annales de l’Institut Henri Poincaré D


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Volume 7, Issue 2, 2020, pp. 233–247
DOI: 10.4171/AIHPD/85

Published online: 2020-06-05

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

Karim A. Adiprasito[1] and Bruno Benedetti[2]

(1) The Hebrew University of Jerusalem, Israel
(2) University of Miami, Coral Gables, USA

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.

Keywords: Discrete quantum gravity, triangulations, collapsibility, discrete finiteness Cheeger theorem, geometric manifolds, bounded geometry, simple homotopy theory

Adiprasito Karim, Benedetti Bruno: A Cheeger-type exponential bound for the number of triangulated manifolds. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 7 (2020), 233-247. doi: 10.4171/AIHPD/85