On Maximal Collections Of Essential Annuli In A Handlebody Or A Compression Body

Topology has developed rapidly since the beginning of the 20th century to thisday, the core direction of which is no doubt the manifold topology, and the case ofthree dimension is much different from others.Firstly, 3-manifolds topology is rich in methods and results. It's because weare familiar with three dimension better than other high dimensions on the one hand;and on the other hand, all kinds of fundamental structures on 3-manifolds are con-sistent nicely with each other for the Hauptvermutung of 3-manifolds is proved to becorrect. Secondly, many classical topological methods (such as Homology) whichhave been proved to be useful in high-dimension cases are not valid in 3-dimensioncase any more. The same phenomenon happens also in 4-manifolds topology, andjust for this reason, there comes about the term of low-dimensional topology. Lastly,from the trend in development of the discipline, the structures in 3-manifolds wouldbe much abundant than ever. Many branches in mathematics, for example the knottheory, have profound connection with 3-manifolds topology, especially in 1970's,W.Thurston historically brought geometry into 3-manifolds which turned over a newleaf of the research of 3-manifolds topology.It is apparently to see from the history that studying special surfaces embed-ded in 3-manifolds is a important method in 3-manifolds topology. Such as the Primedecomposition of 3-manifolds, incompressible surface in 3-manifolds, Heegaard split-ting of 3-manifolds and the JSJ decomposition of 3-manifolds, and so on. All theoriesmentioned above concentrate on certain special surface in 3-manifolds.For a incredible non-Haken 3-manifold M, we study Heegaard splitting and JSJdecomposition structure of M simultaneously. We first splits M open along the Hee-gaard surface to obtain two handlebodies or compression bodies, then it is easy to seethat the JSJ decomposition surface is split into a collection of essential annuli properlyembedded in handlebodies or compression bodies. Starting from this background, wecarry out the research of the maximal collection of essential annuli embedded in ahandlebody or compression body in this paper.In 1999, Rubinstein-Scharlemann finished the classification of pairwise disjoint, nonparallel maximal collection of essential annuli A embedded in the genus 2 handle-body, and they showed that | A |= 1, 2 or at most 3, where |·| denotes the cardinalityof the corresponding set.In this paper, we generalize their results furthermore, and get:(1). For a maximal collection A of pairwise disjoint non-parallel essential annuliin a handlebody of genus n (≥3), 2≤| A |≤4n-5, and the bounds are best possible.(2). For a maximal collection A of pairwise disjoint non-parallel essential non-spanning annuli in a genuine compression body C, 2≤| A |≤4h - b, where h standsfor the total number of 1-handles attached to obtain C, and b denotes the number ofthe components of ??C, and the bounds are best possible.