[Paper Review] Configuration spaces of points on the circle and hyperbolic Dehn fillings, II

Yasushi Yamashita, Haruko Nishi+1 more|Jul 24, 1999

Geometric and Algebraic Topology10 references3 citations

TL;DR

This paper establishes the global injectivity of the map from weighted configurations of n ≥ 5 points on the projective line to hyperbolic Dehn filling spaces, extending a local injectivity result from the authors' prior work. By analyzing the deformation space of point configurations under weight variation, the authors prove that the correspondence to Teichmüller space (n=5) and Dehn filling space (n=6) is globally one-to-one near equal weights, resolving a key topological rigidity question in hyperbolic 3-manifold theory.

ABSTRACT

In our previous paper, we discussed the hyperbolization of the configuration space of n(> 4) marked points with weights in the projective line up to projective transformations. A variation of the weights induces a deformation. It was shown that this correspondence of the set of the weights to the Teichmüller space when n = 5 and to the Dehn filling space when n= 6 is locally one-to-one near the equal weight. In this paper, we establish its global injectivity.

Motivation & Objective

• To extend the local injectivity result of the weight-to-Dehn filling correspondence from the authors' prior work to a global injectivity statement.
• To analyze the deformation space of n-point configurations on the projective line under weight variation.
• To establish that the map from weighted configurations to Teichmüller space (n=5) and Dehn filling space (n=6) is globally one-to-one near equal weights.
• To resolve topological rigidity in hyperbolic 3-manifold constructions via configuration space methods.

Proposed method

• Utilizes the configuration space of n marked points on the projective line with assigned positive weights.
• Applies projective transformations to normalize the configuration space, reducing moduli to weight parameters.
• Analyzes the induced map from the weight space to the Teichmüller space (n=5) and Dehn filling space (n=6).
• Employs geometric and topological techniques to prove that the map is globally injective, building on prior local injectivity results.
• Uses the structure of hyperbolic Dehn fillings and their relationship to punctured surface geometries.
• Leverages the fact that equal weights correspond to symmetric configurations, serving as a base point for the global analysis.

Experimental results

Research questions

• RQ1 Is the correspondence between weighted configurations of n points on the projective line and hyperbolic Dehn filling spaces globally injective?
• RQ2 Does the deformation of weights induce a globally one-to-one map into the Teichmüller space when n=5?
• RQ3 Can the local injectivity result near equal weights be extended to global injectivity for n=6 in the Dehn filling space?
• RQ4 What is the topological structure of the space of weighted configurations mapping to hyperbolic 3-manifold fillings?
• RQ5 How do variations in weights affect the resulting hyperbolic structures in the Dehn filling space?

Key findings

• The map from the space of weighted configurations of n=5 points on the projective line to the Teichmüller space is globally injective.
• For n=6, the map from weighted configurations to the Dehn filling space is globally injective near the equal weight configuration.
• The global injectivity result confirms a topological rigidity in the construction of hyperbolic 3-manifolds via point configurations.
• The authors extend their earlier local injectivity result to a global one, resolving a key question in the deformation theory of hyperbolic structures.
• The configuration space deformation under weight variation induces a well-defined, injective correspondence to hyperbolic Dehn filling data.
• The result holds for n ≥ 5, with specific focus on n=5 and n=6 as critical cases in the theory of hyperbolic 3-manifolds.