Finite Type Invariants of Integral Homology 3-spheres: a Survey

We are now embarrassingly rich in knot and 3-manifold invariants. We have to organize these invariants systematically and find out ways to make use of them. The theory of finite type knot invariants, or Vassiliev invariants, has been very successful in accomplishing the first task. Recently, an analogous theory of finite type invariants of integral homology 3-spheres started to emerge. The analogy is mainly based on the common goal of bringing some order to the multitude of invariants by finding some universal properties they obey. If we think of quantum 3-manifold invariants of Reshetikhin and Turave [44] as non-perturbative ones, their perturbative version [38] is the other source of motivation for this developing theory of finite type invariants of integral homology 3-spheres. Non-perturbative invariants are somehow packed together tightly so that they usually support some very rich algebraic structures. Perturbative invariants, on the other hand, seem to be quite independent with each other. One may see this from Theorem 3.1, the only new result in this paper, which claims that the space of finite type invariants of integral homology 3-spheres is a polynomial algebra. This leads to the speculation that perturbative invariants may contain more geometrical or topological information