Refined quantum invariants for three-manifolds with structure

Abstract

Introduction. Following Witten’s interpretation ([Wi]) of the Jones polynomial ([Jo]) in terms of Topological Quantum Field Theory , Reshetikhin and Turaev ([RT]) and then many others have constructed invariants of 3-manifolds now called Quantum Invariants (see [Tu2] for a detailed exposition, and [Vo] for a survey). The construction of Reshetikhin and Turaev involves representation theory of quantum groups. This point of view gives a deep insight into the algebraic questions related to the subject, however it is not immediately accessible for the beginner. Among these quantum invariants those called the SU(2)-invariants can be obtained easily from the skein theory associated with the Kauffman bracket ([Ka]). This was first observed by Lickorish ([Li1],[Li2],[Li3]) and then systematically studied in [BHMV1]. Section 1 deals with this skein method. Starting with a formal skein theory, we discuss the construction of 3-manifolds invariants, and give the simplest examples. We think that this could be helpful for the beginner and hope that the method will be applied to new examples. Once one has constructed a lot of 3-manifold invariants, the question is to understand their meaning, and this is far from clear at the moment. Let us discuss the example of τ at q=e iπ 8 ([KM]) which corresponds to θ8 in [BHMV1] and [Bl1]. This invariant decomposes as a sum, over all spin structures on the manifold, of spin invariants. Moreover the spin invariant is (a version of) the well known Rochlin invariant. This was first observed by Kirby and Melvin and generalized independently in [KM], [Tu1] and [Bl1]. This example shows that considering refined invariants can help in understanding their geometrical meaning. Section 2 is about cohomological refinements of quantum invariants. According to H. Murakami ([Mu]) τ SU(n) r admits such refinements, for conveniently chosen