A 3-manifold complexity via immersed surfaces

TL;DR

This paper introduces a new invariant called surface-complexity for closed 3-manifolds, measuring their complexity through Dehn surfaces, with properties linking it to existing manifold invariants and decompositions.

Contribution

It defines the surface-complexity invariant, proves its key properties, and relates it to cubulations, triangulations, and other manifold invariants, providing new tools for 3-manifold analysis.

Findings

Surface-complexity is subadditive under connected sum.

It is finite-to-one on P2-irreducible manifolds.

For most P2-irreducible manifolds, it equals the minimal number of cubes in a cubulation.

Abstract

We define an invariant, which we call surface-complexity, of closed 3-manifolds by means of Dehn surfaces. The surface-complexity of a manifold 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 P2-irreducible manifolds. Moreover, for P2-irreducible manifolds, it equals the minimal number of cubes in a cubulation of the manifold, except for the sphere S3, the projective space RP3 and the lens space L41, which have surface-complexity zero. We will also give estimations of the surface-complexity by means of triangulations, Heegaard splittings, surgery presentations and Matveev complexity.