Nested Lemma In Handlebody And Its Application

Casson and Gordon found a new structure-weak reducibility in Heegaard Splitting structure in 1987,which is of great importance for us to explore the properties of 3-manifold by means of Heegaard splitting.In this structure,Boileau and Otal raised Nested Lemma.Scharlemann and Thompson gave a new proof for this lemma depending on Casson and Gordon's results.Enlightened by the form of the results given by the lemma,we try to explore whether there is the same results in general situations.The question is to find out conditions under which the complexity of the intersection of the disk and splitting surface can be reduce to zero.We take compressing and boundary-compressing operation to reduce the complexity by studying the property of incompressible surface in compression body.We find some conditions under which we can reduce the complexity to zero and furthermore we give Nested Lemma in handle-body and general Nested Lemma.Under these conditions,simple closed curve bounding disk in 3-manifold also bounds a disk in sub-manifold,thus we can do further research on the sub-manifold.At last we give applications of these Nested lemmas:in these conditions,the intersections of 3-ball and the splitting have an interesting specialty.