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.
| Comments: | 47 pages. Minor revisions; added an example of a stably metabolic but not metabolic $\mathbb{Z}^P_1$-form, more details about the quadratic tensor product and an equivalent definition of the split norm |
| Subjects: | Geometric Topology (math.GT) |
| MSC classes: | 57R67 |
| Cite as: | arXiv:2404.09189 [math.GT] |
| (or arXiv:2404.09189v2 [math.GT] for this version) | |
| https://doi.org/10.48550/arXiv.2404.09189 arXiv-issued DOI via DataCite |
From: Csaba Nagy [view email]
[v1]
Sun, 14 Apr 2024 08:44:50 UTC (56 KB)
[v2]
Sat, 23 May 2026 11:09:04 UTC (60 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。