Orbifold homeomorphism finiteness based on geometric constraints

TL;DR

This paper proves that under certain geometric constraints, there are only finitely many orbifold types up to homeomorphism, extending known manifold results to orbifolds and linking spectral properties to topological finiteness.

Contribution

It generalizes a manifold finiteness result to orbifolds with bounded curvature, volume, diameter, and isolated singularities, and connects spectral data to orbifold classification.

Findings

Finiteness of orbifold homeomorphism types under geometric bounds

Convergence of orbifold sequences to a limit orbifold

Homeomorphisms preserving orbifold structure via stability theorem

Abstract

We show that any collection of n-dimensional orbifolds with sectional curvature and volume uniformly bounded below, diameter bounded above, and with only isolated singular points contains orbifolds of only finitely many orbifold homeomorphism types. This is a generalization to the orbifold category of a similar result for manifolds proven by Grove, Petersen, and Wu. It follows that any Laplace isospectral collection of orbifolds with sectional curvature uniformly bounded below and having only isolated singular points also contains only finitely many orbifold homeomorphism types. The main steps of the argument are to show that any sequence from the collection has subsequence that converges to an orbifold, and then to show that the homeomorphism between the underlying spaces of the limit orbifold and an orbifold from the subsequence that is guaranteed by Perelman's stability theorem must preserve orbifold structure.