Intersections Of Curves Resulting From Dehn Twist With Disk Systems For A Handlebody

Heegaard splitting is an important branch of 3-manifold topology theory. Basedon the combinatorial structure of 3-manifolds, we can decompose complex geometri-cal objects into some simple blocks by some surfaces and curves in 3-manifolds, suchas Heegaard surfaces, essential disks and essential curves etc. We can study thesesimple objects to learn about the topological properties and geometrical structuresof 3-manifolds. In the dissertation, we mainly study three problems: expressions ofessential disks of a compression body, Casson-Gordon rectangle conditions and inter-sections of curves resulting from Dehn twist with disk systems for a handlebody.The main results of the dissertation are summarized as follows:1. In 2003, Zablow proved that any essential disk in a handlebody can be builtas iterated connected sum of the disks in the standard disk system of a handlebody.In this dissertation we find a standard disk system of a compression body, and proveany essential disk in a compression body can be built as iterated connected sum of thedisks in the standard disk system of a compression body.2. We give a combinatorial condition under which a Heegaard splitting of anorientable 3-manifold with boundary is strongly irreducible.3. We investigate the intersections of curves resulting from Dehn twists on theboundary surface F with standard 3g ? 3 disk systems D for a handlebody H. Wewill give a counting formula for measuring the number of intersections of the result-ing curves with disk systems D for a handlebody H, and estimates for determininglower bounds on Dehn twist by n?closed curves on m?closed curves. In the end,we introduce the branched 1-manifold to simplify the calculation of the number ofintersections of the resulting curves with disk systems D for a handlebody.