Automorphism Groups of Quadratic Modules and Manifolds
type: CreativeWork
description: In this thesis we prove homological stability for both general linear groups of modules over a ring with finite stable rank and unitary groups of quadratic modules over a ring with finite unitary stable rank. In particular, we do not assume the modules and quadratic modules to be well-behaved in any sense: for example, the quadratic form may be singular. This extends results by van der Kallen and Mirzaii--van der Kallen respectively. Combining these results with the machinery introduced by Galatius--Randal-Williams to prove homological stability for moduli spaces of simply-connected manifolds of dimension $2n \geq 6$, we get an extension of their result to the case of virtually polycyclic fundamental groups. We also prove the corresponding result for manifolds equipped with tangential structures. A result on the stable homology groups of moduli spaces of manifolds by Galatius--Randal-Williams enables us to make new computations using our homological stability results. In particular, we compute the abelianisation of the mapping class groups of certain $6$-dimensional manifolds. The first computation considers a manifold built from $\mathbb{R} P^6$ which involves a partial computation of the Adams spectral sequence of the spectrum ${MT}$Pin$^{-}(6)$. For the second computation we consider Spin $6$-manifolds with $\pi_1 \cong \mathbb{Z} / 2^k \mathbb{Z}$ and $\pi_2 = 0$, where the main new ingredient is an~analysis of the Atiyah--Hirzebruch spectral sequence for $MT\mathrm{Spin}(6) \wedge \Sigma^{\infty} B\mathbb{Z}/2^k\mathbb{Z}_+$. Finally, we consider the similar manifolds with more general fundamental groups $G$, where $K_1(\mathbb{Q}[G^{\mathrm{ab}}])$ plays a role.
abstract: In this thesis we prove homological stability for both general linear groups of modules over a ring with finite stable rank and unitary groups of quadratic modules over a ring with finite unitary stable rank. In particular, we do not assume the modules and quadratic modules to be well-behaved in any sense: for example, the quadratic form may be singular. This extends results by van der Kallen and Mirzaii--van der Kallen respectively. Combining these results with the machinery introduced by Galatius--Randal-Williams to prove homological stability for moduli spaces of simply-connected manifolds of dimension $2n \geq 6$, we get an extension of their result to the case of virtually polycyclic fundamental groups. We also prove the corresponding result for manifolds equipped with tangential structures. A result on the stable homology groups of moduli spaces of manifolds by Galatius--Randal-Williams enables us to make new computations using our homological stability results. In particular, we compute the abelianisation of the mapping class groups of certain $6$-dimensional manifolds. The first computation considers a manifold built from $\mathbb{R} P^6$ which involves a partial computation of the Adams spectral sequence of the spectrum ${MT}$Pin$^{-}(6)$. For the second computation we consider Spin $6$-manifolds with $\pi_1 \cong \mathbb{Z} / 2^k \mathbb{Z}$ and $\pi_2 = 0$, where the main new ingredient is an~analysis of the Atiyah--Hirzebruch spectral sequence for $MT\mathrm{Spin}(6) \wedge \Sigma^{\infty} B\mathbb{Z}/2^k\mathbb{Z}_+$. Finally, we consider the similar manifolds with more general fundamental groups $G$, where $K_1(\mathbb{Q}[G^{\mathrm{ab}}])$ plays a role.
author: Friedrich, Nina
language: en
keywords: Algebraic Topolgy, Homological Stability, Stable Homology, Automorphism Groups, Quadratic Modules
license: https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/3eae32fd-9c7a-4b16-920b-8b29df1f9620/download
license: https://www.rioxx.net/licenses/all-rights-reserved/
date published: 2018-07-20
date modified: 2026-07-22T22:24:24Z
provider: Cambridge University
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: University of Cambridge

PropertyValue
property ID: DOI
value: 10.17863/CAM.24264