Generation of Sutured Manifolds

Surgery triads is an important concept in various versions of Floer homology theories of 3-manifolds. If three 3-manifolds form a surgery triad, then the Floer homology of these three manifolds are usually related by an exact triangle. A strategy of proving certain isomorphism theorems for Floer homology is to use exact triangles and naturality. In order to apply this strategy, we often need finite generation results for a class of 3-manifolds through surgery triads. For example, a folklore theorem asserts that all closed oriented connected 3-manifolds are generated by the 3-sphere through surgery triads. In this talk, we will review the above strategy and discuss the proof of the above folklore theorem and its generalization to a class of sutured manifolds.