The Topological Complexity of a Surface

2015/02/10 by Aldo-Hilario Cruz-Cota, Cruz-Cota, Aldo-Hilario
Computer Science · Mathematics · Physics and Astronomy · #30F99 #57M12 #Advanced Mathematical Theories and Applications #Digital Image Processing Techniques #FOS: Mathematics #Geometric Topology (math.GT) #Mathematical Dynamics and Fractals #math.GT #msc:30F99 #msc:57M12

paper · pdf · doi:10.48550/arxiv.1502.03031

12 pages

openalex publication_date 2015/02/10 · arxiv created 2015/04/16 · arxiv updated 2015/04/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p be a branched covering of a Riemann surface to the Riemann sphere ℙ¹, with branching set B ⊂ ℙ¹. We define the complexity of p as infinity, if ℙ¹ ∖ B does not admit a hyperbolic structure, or the product of its degree and the hyperbolic area of ℙ¹ ∖ B, otherwise. The topological complexity of a surface S is defined as the infimum of the set of all complexities of branched coverings M → ℙ¹, where M is a Riemann surface homeomorphic to S. We prove that if S is a connected, closed, orientable surface of genus g, then its topological complexity, C_top(S), is given by: C_top(S)= \ 2π(2g+1) · if g ≥ 1, 6 π · if g=0. .

Related

• The simple complexity of a Riemann surface
• Topological Complexities of Surfaces
• Measuring complexity of curves on surfaces
• Effective topological complexity of orientable-surface groups
• The covering type of closed surfaces and minimal triangulations