{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/276986"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/276986","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"Automorphism Groups of Quadratic Modules and Manifolds","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.","abstract_html":"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 <span class=\"etd-inline-math\">\\mathbb{R} P<sup>6</sup></span> which involves a partial computation of the Adams spectral sequence of the spectrum ${MT}$Pin<span class=\"etd-inline-math\"><sup>-</sup>(6)</span>. For the second computation we consider Spin $6$-manifolds with <span class=\"etd-inline-math\">&pi;<sub>1</sub> \\cong \\mathbb{Z} / 2<sup>k</sup> \\mathbb{Z}</span> and <span class=\"etd-inline-math\">&pi;<sub>2</sub> = 0</span>, where the main new ingredient is an~analysis of the Atiyah--Hirzebruch spectral sequence for <span class=\"etd-inline-math\">MT<span class=\"etd-inline-math-roman\">Spin</span>(6) \\wedge \\Sigma<sup>\\infty</sup> B\\mathbb{Z}/2<sup>k</sup>\\mathbb{Z}<sub>+</sub></span>. Finally, we consider the similar manifolds with more general fundamental groups $G$, where <span class=\"etd-inline-math\">K<sub>1</sub>(\\mathbb{Q}[G<sup><span class=\"etd-inline-math-roman\">ab</span></sup>])</span> plays a role.","abstract_has_math":true,"creators":["Friedrich, Nina"],"institution":"University of Cambridge","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Randal-Williams, Oscar"],"committee_chairs":[],"committee_members":[],"year":2018,"date_issued":"2018-07-20","date_published":"2018-07-20","updated_at":"2026-07-22T22:24:24Z","subjects":["Algebraic Topolgy","Homological Stability","Stable Homology","Automorphism Groups","Quadratic Modules"],"languages":["en"],"rights":[],"rights_urls":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/3eae32fd-9c7a-4b16-920b-8b29df1f9620/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"identifier_entries":[]},"links":{"outbound_url":"https://doi.org/10.17863/CAM.24264","outbound_label":"DOI","outbound_source":"dc:identifier.doi"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Randal-Williams, Oscar"]},{"key":"dc:creator","label":"Author","values":["Friedrich, Nina"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2018-07-20"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cambridge"]},{"key":"dc:relation.isreferencedby.uri","label":"Dc Relation Isreferencedby URI","values":["https://www.repository.cam.ac.uk/handle/1810/276986"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Algebraic Topolgy","Homological Stability","Stable Homology","Automorphism Groups","Quadratic Modules"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/3eae32fd-9c7a-4b16-920b-8b29df1f9620/download","https://www.rioxx.net/licenses/all-rights-reserved/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["10.17863/CAM.24264"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/6f783dd8-e67d-476d-9fad-7bf59620e2b1/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["560905916f86abd371cf6bc6e29b05a9","87eda9de84448d1f82354d60eee3eb5f"]},{"key":"dc:title","label":"Title","values":["Automorphism Groups of Quadratic Modules and Manifolds"]}]}],"canonical_facts":{"dc:contributor.advisor":["Randal-Williams, Oscar"],"dc:creator":["Friedrich, Nina"],"dc:date.issued":["2018-07-20"],"dc:description.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."],"dc:format.checksum.md5":["560905916f86abd371cf6bc6e29b05a9","87eda9de84448d1f82354d60eee3eb5f"],"dc:identifier.doi":["10.17863/CAM.24264"],"dc:identifier.uri":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/6f783dd8-e67d-476d-9fad-7bf59620e2b1/download"],"dc:language":["en"],"dc:publisher.institution":["University of Cambridge"],"dc:relation.isreferencedby.uri":["https://www.repository.cam.ac.uk/handle/1810/276986"],"dc:rights":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/3eae32fd-9c7a-4b16-920b-8b29df1f9620/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"dc:subject":["Algebraic Topolgy","Homological Stability","Stable Homology","Automorphism Groups","Quadratic Modules"],"dc:title":["Automorphism Groups of Quadratic Modules and Manifolds"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["Doctoral"],"dc:type.qualificationname":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-22T22:24:24Z"}