2025/03/08 by Biplab Basak, Sourav Sarkar, Basak, Biplab +1
Computer Science · Mathematics · #05A20 #05C75 #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Primary 05C15 #Secondary 05E45 #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2503.06133
openalex publication_date 2025/03/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This article focuses on a class of properly edge-colored graphs, which arise from topological combinatorics, and investigates their embeddings onto surfaces. Specifically, these graphs are known as the dual graphs of balanced normal pseudomanifolds. We introduce the concept of the balanced genus, which represents the smallest genus of a surface onto which the dual graph of a normal pseudomanifold can embed regularly. As a key result, we establish that for any 3-manifold M that is not a sphere, the balanced genus satisfies the lower bound GM ≥ m+3 , where m is the rank of its fundamental group of M. Furthermore, we prove that a 3-manifold M is homeomorphic to the 3-sphere if and only if its balanced genus GM is at most 3. Similarly, for 4-manifolds, we establish that if M is not homeomorphic to a sphere, then its balanced genus is bounded below by GM ≥ 2χ(M) + 5m + 11 . Moreover, a 4-manifold M is PL homeomorphic to the 4-sphere if and only if its balanced genus satisfies GM ≤ 2χ(M) + 10 . We believe that the balanced genus offers a new perspective in graph theory and combinatorics and will inspire further developments in the field in connection with algebraic combinatorics. To this end, we outline several directions for future research.