Quadratic Functions on Torsion Groups

A quadratic function q on an Abelian groupG is a map, with values in an Abelian group, such that the map b : (x, y) 7→ q(x + y) − q(x) − q(y) is Z-bilinear. Such a map q satisfies q(0) = 0. If, in addition, q satisfies the relation q(nx) = nq(x) for all n ∈ Z and x ∈ G, then q is homogeneous. In general, a quadratic function cannot be recovered from the associated bilinear pairing b. Homogeneous quadratic functions on torsion groups first appeared as quadratic enhancements of the linking pairing on the torsion subgroup of the (2n − 1)-th homology group of an oriented (4n− 1)-manifold. Typically, these quadratic enhancements appear in topology as the manifold is equipped with a framing [BM] [MS] [LL]. They were used as a fundamental ingredient in the classification up to regular homotopy of immersed surfaces in R [Pi]. They were extensively studied from the algebraic viewpoint of Witt and Grothendieck groups, see for instance [Du] [Ka] [La]. However, there is no reason to restrict to homogeneous enhancements of the linking pairing [LW]. It is also convenient to consider possibly degenerate quadratic functions [De1]. The motivation for considering general quadratic functions stems from our work on closed Spin-manifolds of dimension 3 and their finite type invariants [DM]. This paper studies, and gives classification results for, quadratic functions on torsion Abelian groups with values in Q/Z. We prove three results.