Ph.D Thesis

Ph.D StudentKlebanov Anna
SubjectHeegaard and Gauss Diagrams of 3-Manifolds
DepartmentDepartment of Mathematics
Supervisors PROF. Michael Polyak


The paper presents a simple combinatorial method of encoding 3-dimensional manifolds, based on their Heegaard diagrams. The notion of a Gauss diagram of a 3-manifold is introduced. We define the equivalence of Gauss diagrams and prove that there is a one-to-one correspondence between closed oriented 3-manifolds (up to homeomorphism) and Gauss diagrams (up to equivalence). We find the conditions that a Gauss diagram should satisfy so that it would represent a closed manifold and a manifold with boundary. We also show how the fundamental and the first homology groups of the manifold can be computed by means of its Gauss diagram. Some examples of Gauss diagrams of well-known manifolds (such as lens spaces and the Poincare manifold) are given. Gauss diagram invariants of closed orientable 3-manifolds are introduced and partially identified with well-known finite type invariants of 3-manifolds. This gives simple combinatorial formulas for calculation of Casson's invariant of lens spaces and Rokhlin's invariant of lens spaces and their connected sums. The notion of Heegaard diagrams is extended and Θ-Heegaard diagrams are introduced. An analogue of Singer's theorem is proved for Θ-Heegaard diagrams. The notion of Gauss diagrams is extended to Θ-Gauss diagrams and equivalence classes of Θ-Gauss diagrams are defined. We generalize the main results of the previous sections to this case and present some examples.