The Witt groups of extended quadratic forms over Z

We recall the notion of a quadratic form parameter Q over the integers and of 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 (q-1) -connected 2q-manifolds via Q-forms. The algebraic theory of general form parameters and extended quadratic forms has been studied in a variety of far more general settings by authors including Bak, Baues, Ranicki and Schlichting. In this paper we classify all quadratic form parameters Q over the integers, determine the category of quadratic form parameters FP and compute the Witt group functor, \ W₀ FP Ab, Q W₀ (Q), \ where Ab is the category of finitely generated abelian groups and W₀ (Q) is the Witt group of nonsingular Q-forms.