The Witt groups of extended quadratic forms over Z

Abstract

We study quadratic form parameters $Q$ over the integers and extended quadratic forms with values in $Q$, which we call $Q$-forms. Certain form parameters $Q$ appeared in Wall's work on the classification of almost closed $(n-1)$-connected $2n$-manifolds via $Q$-forms. Baues, Ranicki and Schlichting independently developed definitions of extended quadratic forms in more general settings; when restricted to the ring $\mathbb{Z}$, each of those definitions is equivalent to those studied here. In this paper we classify all quadratic form parameters $Q$ over the integers, determine the category of quadratic form parameters $\mathbf{FP}$ and compute the Witt group functor, \[ W_0 \colon \mathbf{FP} \to \mathbf{Ab}, \quad Q \mapsto W_0(Q),\] where $\mathbf{Ab}$ is the category of finitely generated abelian groups and $W_0(Q)$ is the Witt group of nonsingular $Q$-forms.