On finite type 3-manifold invariants III: Manifold weight systems

Abstract

The present paper is a continuation of [11,6] devoted to the study of finite type invariants of integral homology 3-spheres. We introduce the notion of manifold weight systems, and show that type m invariants of integral homology 3-spheres are determined (modulo invariants of type m − 1) by their associated manifold weight systems. In particular, we deduce a vanishing theorem for finite type invariants. We show that the space of manifold weight systems forms a commutative, co-commutative Hopf algebra and that the map from finite type invariants to manifold weight systems is an algebra map. We conclude with better bounds for the graded space of finite type invariants of integral homology 3-spheres.