On abelian quantum invariants of links in 3-manifolds

Abstract

From a finite abelian group G, a quadratic form onG and an element \(c = (c_{1}, \ldots, c_{n})\) in \(G^{n}\), we define a topological invariant \(\tau\) of a pair(M,L) where \(M^{3}\) is a closed oriented 3-manifold and L an oriented, framedn-component link inM. The main result consists in an explicit formula for this invariant, based on a reciprocity formula for Gauss sums, which features a special linking pairing. This pairing depends on both the quadratic form q and the linking pairing of M. A necessary and sufficient condition for the invariant to vanish is described in terms of a characteristic class for this pairing. We also discuss torsion spin\(^c\)-structures and related structures which appear in this context.