Topological collapsings and cylindrical neighborhoods in 3-manifolds

We introduce the concept of topological collapsing as a topological abstraction of polyhedral ones. Then we use this concept to characterize the cylindrical neighborhoods of a closed X in a locally compact separable metric space M such that M - X is a 3-manifold. We also prove the following criterion of existence: X has cylindrical neighborhoods in M iff there is a neighborhood N of X in M which is topologically collapsible onto X respecting Bd(M - X).     

The complex of pant decompositions of a surface

We exhibit a set of edges (moves) and 2-cells (relations) making the complex of pant decompositions on a surface a simply connected complex. Our construction, unlike the previous ones, keeps the arguments concerning the structural transformations independent from those deriving from the action of the mapping class group. The moves and the relations turn out to be supported in subsurfaces with 3g−3+n=1,2 (where g is the genus and n is the number of boundary components), illustrating in this way the so-called Grothendieck principle.     

Compact Stein surfaces with boundary as branched covers of B4

We prove that Stein surfaces with boundary coincide up to orientation preserving diffeomorphisms with simple branched coverings of B ⁴ whose branch set is a positive braided surface. As a consequence, we have that a smooth oriented 3-manifold is Stein fillable iff it has a positive open-book decomposition. Oblatum 24-II-2000 & 30-V-2000?Published online: 16 August 2000     

Representing links in 3-manifolds¶ by branched coverings of S3

We introduce a planar coloured-diagram representation of links in 3-manifolds given as branched coverings of the 3-sphere. We also prove an equivalence theorem based on local moves and the existence of a universal configuration for such representation. As an application we give unified proofs of two different results on existence of fibered links in 3-manifolds. Received: 7 April 1997     

A universal Ribbon Surface in B4

We construct an orientable ribbon surface F B⁴ which is universalin the following sense: any orientable 4-manifold M B⁴ 1-handles 2-handles can be represented as a cover of B⁴ branched overF. 2000 Mathematics Subject Classification 57M12, 57N13, 57Q45.     

Covering homotopy 3-spheres

We prove the following theorem: for any closed orientable 3-manifoldM and any homotopy 3-sphere Σ, there exists a simple 3-fold branched coveringp:M→Σ. We also propose the conjecture that, for any primitive branched coveringp:M→N between orientable 3-manifolds,g(M)≥g(N), whereg denotes the Heegaard genus. By the above mentioned result, the genus 0 case of such conjecture is equivalent to the Poincaré conjecture.