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Normal $4$-pseudomanifolds with a relative 2-skeleton

Biplab Basak, Mangaldeep Saha, Sourav Sarkar

TL;DR

The article advances the combinatorial classification of normal $4$-pseudomanifolds under $g_2$- and $g_3$-optimality by showing that, with exactly one singular vertex, such complexes can be constructed from boundary complexes of $5$-simplices via vertex foldings and connected sums. When two singularities are present, the authors either use a one-vertex suspension of a $g_2$-minimal $3$-pseudomanifold followed by foldings and sums, or rely on edge foldings in combination with connected sums starting from boundary complexes of $5$-simplices. The results provide explicit, constructive frameworks linking $g$-vector optimality to concrete combinatorial operations, with implications for the topological structure and potential generalizations to higher dimensions. Overall, the work offers a detailed, operation-based characterization that connects $g_2$ and $g_3$ invariants to tangible assembly rules for 4-dimensional pseudomanifolds.

Abstract

The study of face-number-related invariants in simplicial complexes is a central topic in combinatorial topology. Among these, the invariant $g_2$ plays a significant role. For a normal $d$-pseudomanifold $K$ ($d \geq 3$), it is known that $g_2(K) \geq g_2(lk(v, K))$ for every vertex $v$. If $K$ has at most two singularities and satisfies $g_2(K) = g_2(lk(t, K))$ for a singular vertex $t$, then $g_3(K) \geq g_3(lk(t,K))$ holds. A normal $d$-pseudomanifold $K$ is called $g_2$- and $g_3$-optimal if $g_2(K) = g_2(lk (t,K))$ and $g_3(K) = g_3(lk (t,K))$ for a singular vertex $t$. In this article, we establish structural results for normal $4$-pseudomanifolds under $g_2$- and $g_3$-optimality conditions. We show that if $K$ is a normal $4$-pseudomanifold with exactly one singular vertex $t$ and is $g_2$- and $g_3$-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 $g_2$- and $g_3$-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 $g_2$- and $g_3$-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.

Normal $4$-pseudomanifolds with a relative 2-skeleton

TL;DR

The article advances the combinatorial classification of normal -pseudomanifolds under - and -optimality by showing that, with exactly one singular vertex, such complexes can be constructed from boundary complexes of -simplices via vertex foldings and connected sums. When two singularities are present, the authors either use a one-vertex suspension of a -minimal -pseudomanifold followed by foldings and sums, or rely on edge foldings in combination with connected sums starting from boundary complexes of -simplices. The results provide explicit, constructive frameworks linking -vector optimality to concrete combinatorial operations, with implications for the topological structure and potential generalizations to higher dimensions. Overall, the work offers a detailed, operation-based characterization that connects and invariants to tangible assembly rules for 4-dimensional pseudomanifolds.

Abstract

The study of face-number-related invariants in simplicial complexes is a central topic in combinatorial topology. Among these, the invariant plays a significant role. For a normal -pseudomanifold (), it is known that for every vertex . If has at most two singularities and satisfies for a singular vertex , then holds. A normal -pseudomanifold is called - and -optimal if and for a singular vertex . In this article, we establish structural results for normal -pseudomanifolds under - and -optimality conditions. We show that if is a normal -pseudomanifold with exactly one singular vertex and is - and -optimal at , then can be obtained from boundary complexes of -simplices through a sequence of operations of types vertex foldings and connected sums. When has exactly two singularities and is - and -optimal at one singular vertex, it is derived from the boundary complexes of -simplices through a sequence of operations of types one-vertex suspensions, vertex foldings, and connected sums. Alternatively, we prove that if has two singular vertices and is - and -optimal at one of them, then it arises from boundary complexes of -simplices through a sequence of operations of types vertex foldings, edge foldings, and connected sums.
Paper Structure (5 sections, 23 theorems, 7 equations, 2 figures)

This paper contains 5 sections, 23 theorems, 7 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Implementation of Combinatorial operations
  4. $g_2$- and $g_3$-optimal normal pseudomanifolds
  5. Future Directions

Key Result

Theorem 1.1

Let $K$ be a normal $4$-pseudomanifold with exactly one singular vertex $t$, and assume that $K$ is $g_2$- and $g_3$-optimal with respect to $t$. Then, $K$ is obtained from the boundary complexes of $5$-simplices by a sequence of operations consisting of vertex foldings and connected sums

Figures (2)

  • Figure 1: $\text{lk} (a,\Delta)$
  • Figure 2: $\text{lk} (v,\Delta)$

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Definition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4: BasakSwartz
  • Definition 3.5: Vertex folding BasakSwartz
  • ...and 27 more