Local Criteria for Triangulating General Manifolds.

Area of Science:

• Topology
• Computational Geometry
• Geometric Modeling

Background:

• Establishing a simplicial complex triangulation for a manifold is a fundamental problem in computational topology.
• Existing methods often rely on differentiability or metric properties, limiting their applicability.
• Algorithms constructing simplicial complexes in local patches require robust triangulation guarantees.

Purpose of the Study:

• To present verifiable criteria for establishing a valid triangulation of a manifold.
• To ensure a given map between a simplicial complex and a manifold is a homeomorphism.
• To provide a general-purpose triangulation guarantee for computational geometry algorithms.

Main Methods:

• Criteria are defined within local coordinate charts of the manifold.
• The approach does not assume a differentiable structure or an explicit metric on the manifold.
• No specific properties, such as the Delaunay property, are assumed for the simplicial complex.

Main Results:

• The proposed criteria ensure that a map H from a simplicial complex to a manifold M is a homeomorphism.
• These criteria are applicable even when the manifold lacks a differentiable structure or metric.
• The criteria are easily verifiable in local settings, suitable for patch-based construction algorithms.

Conclusions:

• The presented criteria offer a robust method for manifold triangulation.
• This work provides a theoretical guarantee for algorithms that construct simplicial complexes.
• The criteria are expected to be broadly useful in computational topology and geometry.