Algorithms and topological invariants for dynamic systems. III. Algorithms for Recognition and Classification of 2-Dimensional Surfaces

TL;DR

This paper develops algorithms and invariants to identify and classify 2-dimensional surfaces and their topological features within simplicial and CW-complex structures, advancing topological recognition methods.

Contribution

It introduces new algorithms for recognizing 2-manifolds and determining their topological properties within complex structures, building on prior differential and discrete topology work.

Findings

Algorithms successfully recognize 2-manifolds in simplicial complexes.

Determination of topological invariants like genus and orientability.

Effective classification of 2-manifolds based on topological invariants.

Abstract

We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part (arXiv:2501.15657), we discused basic concepts of diferential topology. In the second part (arXiv:2502.00506 ) we discused the main discrete topological structures used in the topological theory of dynamic systems. In third part we construct algorithms that allow us recognise 2-manifolds and 3-manifolds in simplicial complexes and regular CW-complex and detreminate topological type for 2-manifolds (connectivity, type of orientability, genus, number of boundary component).