The Witt groups of extended quadratic forms over Z

2024/04/14 by Diarmuid Crowley, Crowley, Diarmuid, Csaba Nagy +1
Mathematics · #57R67 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2404.09189

openalex publication_date 2024/04/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 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 FP and compute the Witt group functor, W₀ \colon 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.

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