Towards an Elementary Theory of Finite Type Invariants of Integral Homology Spheres

Following Ohtsuki, Garoufalidis, and Habegger we provide an elementary introduction to nite type invariants of integral homology spheres, culminating with a proof of the upper bound for the magnitude of the space I of such invariants in terms of the space A(;) of oriented trivalent graphs modulo the so-called AS and IHX relations. We raise the issue of \the fundamental theorem" for nite type invariants of integral homology spheres, which says that A(;) is also a lower bound for I (and hence, up to the diierence between a ltered space and a graded space, the two are equal). There are several constructive but transcendental proofs of the fundamental theorem, and we underline a few problems whose solution may yield a direct topological proof of that theorem. Contents 1. Introduction 2 1.1. Stories 2 1.2. Plan of the paper 4 1.3. Finite type invariants of integral homology spheres, the deenition 5 1.4. Acknowledgement 6 2. The case of knots 6