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Heegaard diagrams and applications Li, Zhongmou

Abstract

The main objective of this thesis is to study Heegaard diagrams and their applications. First, we investigate Heegaard diagrams of closed 3-manifolds and introduce the circle and chord presentation for a connected, closed 3-manifold. The equivalence problem for Heegaard diagrams after connected sum moves and Dehn twists will be investigated. Presentations will be used to detect reducible Heegaard diagrams and homeomorphic 3-manifolds. We also investigate Heegaard diagrams of the 3-sphere. The main result of this part is that if two Heegaard diagrams of the 3-sphere have the same genus, then there is a sequence of connected sum moves and Dehn twists to pass from one to the other. If we use connected sum moves only, Heegaard curves can be changed to primitive curves and if we use Dehn twists only Heegaard curves can be brought into a simple position. Finally, we construct an immersion of a compact, orientable, connected 3-manifold with non-empty boundary into R³ with at most double and triple points as singularities. Further, we prove that if the boundary of the 3-manifold consists of 2-spheres and the 3-manifold can immerse into R³ with only double points as singularities, then the 3- manifold must be a punctured 3-sphere or a punctured (S¹ x S²)# • • • #(S¹ x S²).

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