An Invariant of Integral Homology 3-Spheres Which Is Universal For All Finite Type Invariants

Abstract

In [LMO] a 3-manifold invariant $Ω(M)$ is constructed using a modification of the Kontsevich integral and the Kirby calculus. The invariant $Ω$ takes values in a graded Hopf algebra of Feynman 3-valent graphs. Here we show that for homology 3-spheres the invariant $Ω$ is {\em universal} for all finite type invariants, i.e. $Ω_n$ is an invariant of order $3n$ which dominates all other invariants of the same order. This shows that the set of finite type invariants of homology 3-spheres is equivalent to the Hopf algebra of Feynman 3-valent graphs. Some corollaries are discussed. A theory of groups of homology 3-spheres, similar to Gusarov's theory for knots, is presented.