Abstract

Let M be a connected, compact 3-manifold with a fixed triangulation $\cal T$. There is a linear system, called the system of matching equations, associated with $\cal T$. A normal surface, a properly immersed surface which intersects each tetrahedron of $\cal T$ nicely, corresponds to a non-negative integral solution to the system of matching equations. A non-negative integral solution to the system of matching equations is called admissible is there is a normal surface corresponding to it. The system of normal equations is obtained by adding the normalizing equation to the system of matching equations. The space of solutions to the system normal equations is a convex cell in $R\sp{n}$ and is called the projective solution space of M associated with $\cal T$.

In this thesis we give necessary and sufficient conditions to determine if a non-negative integral solution to the system of matching equations is admissible. We also study the projective solution spaces of 3-manifolds which fibers over $S\sp1$. At last we characterize certain kind incompressible surfaces properly embedded in F $\times$ I, where F is an orientable, closed surface.