Research Paper#Mathematics, Topology, Group Theory, Descriptive Set Theory🔬 ResearchAnalyzed: Jan 3, 2026 09:20
Borel Complexity of Manifold and Group Classification
Published:Dec 31, 2025 17:45
•1 min read
•ArXiv
Analysis
This paper investigates the classification of manifolds and discrete subgroups of Lie groups using descriptive set theory, specifically focusing on Borel complexity. It establishes the complexity of homeomorphism problems for various manifold types and the conjugacy/isometry relations for groups. The foundational nature of the work and the complexity computations for fundamental classes of manifolds are significant. The paper's findings have implications for the possibility of assigning numerical invariants to these geometric objects.
Key Takeaways
- •Applies descriptive set theory to classify manifolds and groups.
- •Determines Borel complexity of homeomorphism and conjugacy/isometry relations.
- •Provides insights into the possibility of numerical invariants for geometric objects.
- •Establishes the complexity of classifying complete hyperbolic n-manifolds with finitely generated fundamental groups.
Reference
“The paper shows that the homeomorphism problem for compact topological n-manifolds is Borel equivalent to equality on natural numbers, while the homeomorphism problem for noncompact topological 2-manifolds is of maximal complexity.”