2025/05/06 by Biplab Basak, Basak, Biplab, Sourav Sarkar +2
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2505.03413
openalex publication_date 2025/05/06 · openalex created_date 2025/10/16 · openalex updated_date 2026/07/28
The study of face-number-related invariants in simplicial complexes is a central topic in combinatorial topology. Among these, the invariant g2 plays a significant role. For a normal d-pseudomanifold K (d ≥ 3), it is known that g2(K) ≥ g2(lk(v, K)) for every vertex v. If K has at most two singularities and satisfies g2(K) = g2(lk(t, K)) for a singular vertex t, then g3(K) ≥ g3(lk(t,K)) holds. A normal d-pseudomanifold K is called g2- and g3-optimal if g2(K) = g2(lk (t,K)) and g3(K) = g3(lk (t,K)) for a singular vertex t. In this article, we establish structural results for normal 4-pseudomanifolds under g2- and g3-optimality conditions. We show that if K is a normal 4-pseudomanifold with exactly one singular vertex t and is g2- and g3-optimal at t, then K can be obtained from boundary complexes of 5-simplices through a sequence of operations of types vertex foldings and connected sums. When K has exactly two singularities and is g2- and g3-optimal at one singular vertex, it is derived from the boundary complexes of 4-simplices through a sequence of operations of types one-vertex suspensions, vertex foldings, and connected sums. Alternatively, we prove that if K has two singular vertices and is g2- and g3-optimal at one of them, then it arises from boundary complexes of 5-simplices through a sequence of operations of types vertex foldings, edge foldings, and connected sums.