A complexity of compact 3-manifold via immersed surfaces

TL;DR

The paper defines surface-complexity $sc(M)$ for connected compact 3-manifolds via quasi-filling Dehn surfaces, and develops a robust framework linking immersed surface models to cubulations and classical complexity measures. It proves finiteness, subadditivity under connected sums, and, for $\\mathbb{P}^2$-irreducible, boundary-irreducible manifolds without certain obstructions, an equality $sc(M)$ with the minimal number of cubes in an ideal cubulation (except for a short list of zero cases). It also shows that every manifold admits a minimal filling Dehn surface, and provides practical estimations by relating $sc(M)$ to ideal triangulations and Matveev complexity, yielding useful bounds and reconstruction procedures. Overall, the work situates surface-complexity as a natural, computable invariant with deep connections to cubulations, triangulations, and established 3-manifold complexity frameworks, offering a path toward classification and quantitative comparisons across manifolds.

Abstract

We define an invariant, which we call surface-complexity, of compact 3-manifolds by means of Dehn surfaces. The surface-complexity is a natural number measuring how much the manifold is complicated. We prove that it fulfils interesting properties: it is subadditive under connected sum and finite-to-one on $\mathbb{P}^2$-irreducible and boundary-irreducible manifolds without essential annuli and Möbius strips. Moreover, for these manifolds, it equals the minimal number of cubes in a cubulation of the manifold, except for the sphere, the ball, the projective space and the lens space $\mathbb{L}_{4,1}$, which have surface-complexity zero. We will also give estimations of the surface-complexity by means of ideal triangulations and Matveev complexity.

Figures (11)

• Figure 1: Neighbourhoods of points (marked by thick dots) of a Dehn surface.
• Figure 2: A cubulation of the 3-dimensional torus $\mathbb{S}^1\times\mathbb{S}^1\times\mathbb{S}^1$ with two cubes (the identification of each pair of faces is the obvious one, i.e. the one without twists).
• Figure 3: A cube with small tetrahedra removed near the vertices.
• Figure 4: Local behaviour of duality.
• Figure 5: Construction of a filling Dehn sphere from an ideal triangulation.

Theorems & Definitions (17)

• Remark 1
• Remark 2
• Theorem 3
• proof
• Remark 4
• Definition 5
• Theorem 6
• proof
• Corollary 7
• Corollary 8