Sweepouts of amalgamated 3–manifolds

Abstract

We show that if two 3-manifolds with toroidal boundary are glued via a "sufficiently complicated" map then every Heegaard splitting of the resulting 3-manifold is weakly reducible. Additionally, suppose X ∪ F Y is a manifold obtained by gluing X and Y , two connected small manifolds with incompressible boundary, along a closed surface F . Then the following inequality on genera is obtained:Both results follow from a new technique to simplify the intersection between an incompressible surface and a strongly irreducible Heegaard splitting.

On non-compact Heegaard splittings

“…It is, perhaps, well-known. It appears in similar versions as Lemma 5.2 in [1] and as a remark following Definition 2.1 in [13]. Proof Let y U [ y S y V be the absolute Heegaard splitting for N obtained by including Á.V \ B/ into U for each component B @N which intersects S .…”

“…Y between complexes is proper if the preimage of each compact set is compact. If X is a surface and Y is 3-manifold, is a proper embedding if, in addition to being proper and an embedding, 1 . @Y / D @X .…”

“…Y is proper if it is proper as a map. If X is a surface and Y is a 3-manifold we also require that 1 . @Y / D @.X I /.…”

“…Remark There are many other similar constructions of 3-manifolds with infinitely generated fundamental group which have non-stabilized splittings. By allowing arbitrary glueing maps between boundary tori of the compact pieces, one can use a theorem of Bachman, Schleimer, and Sedgewick [1] to show that the amalgamated splittings are not stabilized. We do not pursue this route further in this paper.…”

“…The only slides we performed were of the disc G over other discs, and since intersected G , G was not a disc of fr K nC1 [ fr K q . Furthermore, since the discs of 1 show up as spots on S 1 it is easy to arrange these slides to be relative to 1 . Thus, these handle-slides are of the sort allowed in sequences in L. Let l denote the sequence of these handle-slides followed by the slide-move (M2) where we add the disc D 0 to 1 .…”