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Small Triangulations of Simply Connected 4-Manifolds

Jonathan Spreer, Lucy Tobin

TL;DR

The paper addresses the problem of triangulation complexity for simply connected 4-manifolds by constructing explicit generalised triangulations of all connected sums of $\mathbb{C}P^2$ and $S^2\times S^2$ with $2\beta_2+2$ pentachora and arguing these are minimal for their topological types. The authors introduce four infinite families of triangulations, provide a robust connected-sum construction compatible with triangulations, and identify their PL-homeomorphism types via Pachner moves, aided by computer-aided verifications. A central conjecture is that the complexity satisfies $c(M) \ge 2\beta_2(M)+2$, and the work advances toward proving minimality in many simply connected 4-manifolds (while noting K3 remains outside current results). The methodology combines careful combinatorial constructions (bow and hook units), a systematic use of Pachner moves that commute with gluings, and a computational framework to certify PL-equivalence, offering a concrete path to minimal triangulations in this dimension. Overall, the paper provides explicit, scalable triangulations for main topological types in 4-manifold theory and a practical, computer-assisted approach to proving PL-homeomorphism, with potential implications for broader triangulation complexity questions and K3-related cases.

Abstract

We present small triangulations of all connected sums of $\mathbb{CP}^2$ and $S^2 \times S^2$ with the standard piecewise linear structure. Our triangulations have $2β_2+2$ pentachora, where $β_2$ is the second Betti number of the manifold. By a conjecture of the authors and, independently, Burke, these triangulations have the smallest possible number of pentachora for their respective topological types.

Small Triangulations of Simply Connected 4-Manifolds

TL;DR

The paper addresses the problem of triangulation complexity for simply connected 4-manifolds by constructing explicit generalised triangulations of all connected sums of and with pentachora and arguing these are minimal for their topological types. The authors introduce four infinite families of triangulations, provide a robust connected-sum construction compatible with triangulations, and identify their PL-homeomorphism types via Pachner moves, aided by computer-aided verifications. A central conjecture is that the complexity satisfies , and the work advances toward proving minimality in many simply connected 4-manifolds (while noting K3 remains outside current results). The methodology combines careful combinatorial constructions (bow and hook units), a systematic use of Pachner moves that commute with gluings, and a computational framework to certify PL-equivalence, offering a concrete path to minimal triangulations in this dimension. Overall, the paper provides explicit, scalable triangulations for main topological types in 4-manifold theory and a practical, computer-assisted approach to proving PL-homeomorphism, with potential implications for broader triangulation complexity questions and K3-related cases.

Abstract

We present small triangulations of all connected sums of and with the standard piecewise linear structure. Our triangulations have pentachora, where is the second Betti number of the manifold. By a conjecture of the authors and, independently, Burke, these triangulations have the smallest possible number of pentachora for their respective topological types.

Paper Structure

This paper contains 15 sections, 20 theorems, 7 equations, 6 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Background
  3. 4-Manifolds
  4. Triangulations
  5. Elementary Moves
  6. Setup
  7. The Connected Sums of $\mathbb{CP}^2$ and $S^2 \times S^2$
  8. Connected Sums of 4-Manifold Triangulations
  9. The triangulations $\mathcal{P}(k)$, $\mathcal{E}(k)$, $\mathcal{A}(k)$ and $\mathcal{D}(k,l)$
  10. Identifying the PL Types of $\mathcal{P}(k)$, $\mathcal{E}(k)$, $\mathcal{A}(k)$ and $\mathcal{D}(k,l)$
  11. Computational Methods
  12. The Basic Algorithm
  13. The Algorithm with Simplification
  14. Sequences of Elementary Moves
  15. Code

Key Result

Theorem 1.1

Let $\mathcal{M}$ be a simply connected $4$-manifold obtained from connected sums of PL standard summands $\mathbb{C}P^2$ and $S^2 \times S^2$ in any orientation, then $\mathcal{M}$ can be triangulated using $2 \beta_2(\mathcal{M}) + 2$ pentachora.

Theorems & Definitions (35)

  • Theorem 1.1
  • Conjecture 1.2
  • Theorem 2.1: Freedman Freedman1982
  • Theorem 2.2: Donaldson Donaldson1983
  • Theorem 2.3: Rokhlin Rokhlin1952
  • Conjecture 2.4: The 11/8 Conjecture
  • Lemma 2.5: Milnor2013
  • Theorem 2.6: Milnor2013
  • Corollary 2.7
  • Definition 2.8
  • ...and 25 more