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Vertex-Minimal Triangulation of Complexes with Homology

Jon V. Kogan

TL;DR

Problem: determine the minimal vertex count for pure d-dimensional simplicial complexes with nontrivial H_k, under both unrestricted and strongly connected regimes. Approach: derive exact lower bounds using nerve arguments, Mayer–Vietoris, and explicit constructions (MH_{d,k} for the pure case and MS_{d,k} for the strongly connected case); extend to a general relative strong connectivity theorem. Key contributions: sharp, tight bounds with constructions, a general relative framework, and topological implications for cohomology operations and collapsibility. Significance: provides precise extremal benchmarks for vertex counts in combinatorial topology and informs related structures such as cup products, Steenrod operations, and growth processes in d-dimensional complexes.

Abstract

For a given pair of numbers $(d,k)$, we establish the minimal number of vertices in pure $d$-dimensional simplicial complexes with non-trivial homology in dimension $k$. Furthermore, we solve the problem under the additional constraint of strong connectivity with respect to any intermediate dimension.

Vertex-Minimal Triangulation of Complexes with Homology

TL;DR

Problem: determine the minimal vertex count for pure d-dimensional simplicial complexes with nontrivial H_k, under both unrestricted and strongly connected regimes. Approach: derive exact lower bounds using nerve arguments, Mayer–Vietoris, and explicit constructions (MH_{d,k} for the pure case and MS_{d,k} for the strongly connected case); extend to a general relative strong connectivity theorem. Key contributions: sharp, tight bounds with constructions, a general relative framework, and topological implications for cohomology operations and collapsibility. Significance: provides precise extremal benchmarks for vertex counts in combinatorial topology and informs related structures such as cup products, Steenrod operations, and growth processes in d-dimensional complexes.

Abstract

For a given pair of numbers , we establish the minimal number of vertices in pure -dimensional simplicial complexes with non-trivial homology in dimension . Furthermore, we solve the problem under the additional constraint of strong connectivity with respect to any intermediate dimension.
Paper Structure (6 sections, 7 theorems, 12 equations, 2 figures)

This paper contains 6 sections, 7 theorems, 12 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Pure Dimensional Case
  4. Strongly Connected Case
  5. Relative Strong Connectivity
  6. Discussion

Key Result

Theorem 1.1

Let $0\le k\le d$. Any pure $d$-dimensional simplicial complex with nontrivial $H_k$ has at least $\left\lceil \frac{(d+1)(k+2)}{k+1} \right\rceil$ vertices, and this bound is tight.

Figures (2)

  • Figure 1: The first examples for when $k=1$.
  • Figure 2: The first examples for when $k=2$

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Claim 2.4: from the preprint kogan24
  • proof
  • Example 2.5
  • Theorem 2.6: The Mayer–Vietoris sequence,AT
  • ...and 17 more