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• W3143689636 abstract "ABSTRA~~. In a recent paper we used Cerf theory to compare strongly irreducible Heegaard splittings of the same closed irreducible orientable 3-manifold. This captures all irreducible splittings of non-Haken 3-manifolds. One application is a solution to the stabilization problem for such splittings: If p < q are the genera of two s11littings, then there is a common stabilization of genus 51, + 8q - 9. Here xve show how to obtain similar results even ~vhen the 3manifold has boundary. In this paper, all manifolds are assumed to be compact and orientable A compresszon body H is constructed by adding 2-handles to a (connected surface) x I along a collection of disjoint simple closed curves on (surface) x (01, and capping off any resulting 2-sphere boundary components with 3-balls. 'The component (surface) x (1) of dH is denoted a+H and the surface aH - a,.H is denoted d-H. If d-H = 0 then H is a handlebody. If H = a+H x I then H is called a trzvzal compression body. The cores of the 2-handles defining H, extended vertically down through (d+H) x I,are called a definzng sef of 2-disks for H. There is a dual picture: A spzne for H is a properly imbedded 1-complex E such that H deformation retracts to E! U 8-H. Such a spine can be constructed from a set of defining disks: The arc co-cores of the 2-handles, with the arc ends that lie on 2-spheres coned to the centers of the 3-balls and the other ends extended clown to d-H, are the edges of a spine. The retraction restricts to a map d.bH -+ E U d-H whose mapping cylinder is itself homeomorphic to H. A spannzng annulus in H is a properly imbedded annulus A with one end on each of a* H. For A a spanning annulus, let d* A = A n d+ H. A Heegaard splzttzng M := AUpB of a 3-manifold consists of an orientable surface P in hl-, together with two compression bodies A and B into which P divides 144. P itself is called the splitting surface. The genus of A Up B is defined to be the genus of P. A Heegaard splitting of M can also be viewed as a handle struciure on 114, in which the 1-handles are the co-cores of the defining 2-handles of A and the 2-handles are the defining 2-handles of B. A stabzlzxatzon of A Up B is the Heegaard splitting obtained by adding to A a regular neighborhood of a proper arc in B which is parallel in B to an arc in P. A stabilization has genus one larger" @default.
• W3143689636 created "2021-04-13" @default.
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• W3143689636 date "1998-01-01" @default.
• W3143689636 modified "2023-09-26" @default.
• W3143689636 title "COMPARING HEEGAARD SPLITTINGS -THE BOUNDED CASE" @default.
• W3143689636 hasPublicationYear "1998" @default.
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