Narrow Results

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 L_4,1, 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.

A Dehn sphere Σ [C. D. Papakyriakopoulos, On Dehn's Lemma and the asphericity of knots, Ann. Math. (2) 66 (1957) 1–26] in a closed 3-manifold M is a sphere immersed in M with only double curve and triple point singularities. The Dehn sphere Σ ⊂ M lifts to M × I, where I is an interval, if there exists an embedded sphere in M × I that projects onto Σ. Every closed 3-manifold has a filling Dehn sphere [J. M. Montesinos-Amilibia, Representing 3-Manifold by Dehn Spheres, Contribuciones Matemáticas: Homenaje a Joaquín Arregui Fernández (Editorial Complutense, 2000), pp. 239–247], i.e. a Dehn sphere that defines a cell decomposition of M. In [R. Vigara, Representación de 3-variedades por esferas de Dehn rellenantes, Ph.D. Thesis, UNED, Madrid (2006)], it is shown that every closed 3-manifold M has a filling Dehn sphere that lifts to M × I. In this paper it is proved that every closed 3-manifold has a filling Dehn sphere that does not lift to M × I. These results solve a question of Roger Fenn.

We prove that any knot or link in any $3$-manifold can be nicely decomposed (split) by a filling Dehn sphere. This has interesting consequences in the study of branched coverings over knots and links. We give an algorithm for computing Johansson diagrams of filling Dehn surfaces out from coverings of $3$-manifolds branched over knots or links.

For immersed surfaces in the four-space, we have a generating set of the Swenton–Hughes–Kim–Miller spatial moves that relate singular banded diagrams of ambient isotopic immersions of those surfaces. We also have Yoshikawa–Kamada–Kawauchi–Kim–Lee planar moves that relate marked graph diagrams of ambient isotopic immersions of those surfaces. One can ask if the former moves form a minimal set and if the latter moves form a generating set. In this paper, we derive a minimal generating set of spatial moves for diagrams of surfaces immersed in the four-space, which translates into a generating set of planar moves. We also show that the complements of two equivalent immersed surfaces can be transformed one another by a Kirby calculus not requiring the 1-1-handle or 2-1-handle slides. We also discuss the fundamental group of the immersed surface-link complement in the four-space and a quandle coloring invariant of an oriented immersed surface-link.