Automorphism Groups of Quadratic Modules and Manifolds

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} P⁶ 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 π₁ \cong \mathbb{Z} / 2^k \mathbb{Z} and π₂ = 0, where the main new ingredient is an~analysis of the Atiyah--Hirzebruch spectral sequence for MTSpin(6) \wedge \Sigma^\infty B\mathbb{Z}/2^k\mathbb{Z}₊. Finally, we consider the similar manifolds with more general fundamental groups $G$, where K₁(\mathbb{Q}[G^ab]) plays a role.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2018

Author dc:creator

• Friedrich, Nina

Advisor dc:contributor.advisor

• Randal-Williams, Oscar

• Algebraic Topolgy
• Homological Stability
• Stable Homology
• Automorphism Groups
• Quadratic Modules

Licence

• https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/3eae32fd-9c7a-4b16-920b-8b29df1f9620/download
• https://www.rioxx.net/licenses/all-rights-reserved/

Language dc:language

en

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.24264

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/276986