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Non-orientable 3-manifolds of surface-complexity one

Gennaro Amendola

TL;DR

The paper classifies closed non-orientable $ P^2$-irreducible 3-manifolds of surface-complexity one, equivalently those with cubic-complexity one, by tying surface-complexity to cubic- and Matveev-complexities. It shows that any such manifold must admit a cubulation with a single cube, and uses this to bound the Matveev complexity to at most $6$, reducing the search to a finite list: the four flat manifolds and the torus bundle with monodromy $(1110)$. A constructive analysis via block-based triangulations from one-cube cubulations demonstrates that the four flat manifolds indeed realize SC=1, while an edge-valence argument excludes the torus bundle as a one-cube cubulation, thereby proving the classification. The result highlights the simplicity of the non-orientable SC=1 landscape and provides a framework for computer-aided extensions to higher complexities.

Abstract

We classify all closed non-orientable $\mathbb{P}^2$-irreducible 3-manifolds obtained by identifying the faces of a cube. These turn out to be the closed non-orientable $\mathbb{P}^2$-irreducible 3-manifolds with surface-complexity one. We show that they are the four flat ones.

Non-orientable 3-manifolds of surface-complexity one

TL;DR

The paper classifies closed non-orientable -irreducible 3-manifolds of surface-complexity one, equivalently those with cubic-complexity one, by tying surface-complexity to cubic- and Matveev-complexities. It shows that any such manifold must admit a cubulation with a single cube, and uses this to bound the Matveev complexity to at most , reducing the search to a finite list: the four flat manifolds and the torus bundle with monodromy . A constructive analysis via block-based triangulations from one-cube cubulations demonstrates that the four flat manifolds indeed realize SC=1, while an edge-valence argument excludes the torus bundle as a one-cube cubulation, thereby proving the classification. The result highlights the simplicity of the non-orientable SC=1 landscape and provides a framework for computer-aided extensions to higher complexities.

Abstract

We classify all closed non-orientable -irreducible 3-manifolds obtained by identifying the faces of a cube. These turn out to be the closed non-orientable -irreducible 3-manifolds with surface-complexity one. We show that they are the four flat ones.
Paper Structure (5 sections, 3 theorems, 4 figures, 3 tables)

This paper contains 5 sections, 3 theorems, 4 figures, 3 tables.

Table of Contents

  1. Definitions
  2. The classification
  3. Matveev complexity and triangulations
  4. From cubulations to triangulations
  5. Proof of Theorem \ref{['theorem:sc1_nonor']}

Key Result

Theorem 3

The surface-complexity of a (connected and closed) $\mathbb{P}^2$-irreducible 3-manifold, different from the sphere $S^3$, the projective space $\mathbb{RP}^3$ and the Lens space $L_{4,1}$, is equal to the cubic-complexity of $M$. The three manifolds $S^3$, $\mathbb{RP}^3$ and $L_{4,1}$ have surface

Figures (4)

  • Figure 1: A cubulation of the 3-dimensional torus $S^1\times S^1\times S^1$ with one cube (the letters show that the identification of each pair of opposite faces is the obvious one, i.e. the one without twists and reflections).
  • Figure 2: Neighbourhoods of points (marked by thick dots) of a Dehn surface (the lines of double points are drawn thick).
  • Figure 3: The diagonal pattern of the $5$-tetrahedron block.
  • Figure 4: The diagonal pattern if one (a), two (b) or three (c) pairs of diagonals do not match.

Theorems & Definitions (9)

  • Definition 1
  • Definition 2
  • Theorem 3
  • Theorem 4
  • Remark 5
  • Lemma 6
  • proof
  • proof
  • Remark 7