3-Manifolds of S1-Category Three

I study 3-manifold theory, which is a fascinating research area in topology. Many new ideas and techniques were introduced during these years, which makes it an active and fast developing subject. It is one of the most fruitful branches of today's mathematics and with the solution of the Poincare conjecture, it is getting more attention. This dissertation is motivated by results about categorical properties for 3-manifolds. This can be rephrased as the study of 3-manifolds which can be covered by certain sets satisfying some homotopy properties. A special case is the problem of classifying 3-manifolds that can be covered by three simple S1-contractible subsets. S1-contractible subsets are subsets of a 3-manifold M3 that can be deformed into a circle in M3. In this thesis, I consider more geometric subsets with this property, namely subsets are homeomorphic to 3-balls, solid tori and solid Klein bottles. The main result is a classication of all closed 3-manifolds that can be obtained as a union of three solid Klein bottles.