| The theory of Heegaard splittings is an important part of3-manifold topology. After Casson-Gordon introducing the idea of weakly reducible Heegaard splittings in1987, the theory has been developed rapidly. In2001. Hempel applied the curve complex theory which was introduced by Harvey in1970s and developed by many others hereafter, to introduce the distances of Heegaard splittings to deal with some problems in the theory of Heegaard splittings and related topics. Since then, much progress has been made in the field, including the rich results regarding to amalgamations and stabilizations of Heegaard splittings.In the thesis, we generalize the concepts of an edge path, as well as the distance, of two vertices in the curve complex of a surface to that of an (i+1)-path. as well as i-distance, between two i-simplices in the curve complex and obtain some properties about Pi-conncetedness. For a reducible Heegaard splitting, we consider a maximal prime connected sum decomposition of the Heegaard splitting and describe the relation between the dimension of the intersection of the two associated disk sub-complexes and the number of various factors in the decomposition. In addition, we obtain some properties on planar busting curves on the boundary of a handlebody. The main results are as follows1. Let Sg be the orient able closed surface of genus g≥2. For0≤i≤3g-5.(Sg) is Pi-connected and its i-diameter is infinite. Moreover, for1≤i≤3g-5. the relationship between the i-distance and the (i-1)-distance is given.2. Let V∪S W be a reducible Heegaard splitting of genus g≥2with a maximal prime connected sum decomposition, and p, q the numbers of irreducible factors and the genus1factors of S2×S1in the decomposition. Let CV and CW be the associated disk sub-complexes of C(S). Then dim(CV∩CW)=2p+3q-4.3. There exist planar-busting curves on the boundary of a handlebody of genus n≥2. Moreover, a pants-busting curve must also be annulus-busting if the genus of the handlebodv is at least3. |
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