{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/399250"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/399250","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"Compact G2-orbifolds via Twisted Connected Sums and Associative 3-folds","abstract":"In this thesis, we study the possibility of extending the well-established construction of compact 7-manifolds carrying an In this thesis, we study the possibility of extending the well-established construction of compact $7$-manifolds carrying an irreducible torsion-free $G_2$-structure, known as the \\emph{Twisted Connected Sum}, to the setting of $7$-orbifolds, spaces locally modeled on $\\mathbb{R}^7/\\Gamma$, quotients of $\\mathbb{R}^7$ by finite subgroups $\\Gamma$ of the group $G_2$. Our work extends previous results by Alexei Kovalev (\\cite{Kov1}) and Dominic Joyce (\\cite{Joy1}) on the existence of torsion-free $G_2$-structures and establishes a topological criterion for the irreducibility of such structures in the case of orbifolds. The strategy for the existence part is to lift the problem locally to a $\\Gamma$-invariant problem on a manifold. For irreducibility, the strategy is to adapt a criterion due to Joyce by considering a topological invariant for orbifolds called the \\emph{orbifold fundamental group}. We also investigate the irreducibility of a number of examples found in the literature, prove that the irreducibility of a global quotient of a $G_2$-manifolds is equivalent to the irreducibility of the manifold, and construct a few dozen examples by using weighted projective spaces as the building blocks of the twisted connected sum. Another result in the thesis is a classification of associative $3$-folds in product $G_2$-manifolds of the form $X\\times T^3$, and related $G_2$ orbifolds of the form $(X\\times T^3)/\\mathbb{Z}_2^2$ where $X$ is a hyper-Kähler $K3$ surface. The defining condition for this class is that the derivative of the torus projection has constant rank. We prove that under these assumptions, up to isometry of the ambient $G_2$ $7$-fold, this class consists of associative $3$-folds which are given by the quotients of either products of the form $\\Sigma\\times\\gamma$, where $\\Sigma$ is a complex curve in $X$ and $\\gamma$ is an appropriately chosen embedded circle in $T^3$, or by $3$-tori, $\\{x_0\\}\\times T^3$, where $x_0\\in X$. Another result in the thesis is a classification of associative $3$-folds in product $G_2$-manifolds of the form $X\\times T^3$, and related $G_2$ orbifolds of the form $(X\\times T^3)/\\mathbb{Z}_2^2$ where $X$ is a hyper-Kähler $K3$ surface. The defining condition for this class is that the derivative of the torus projection has constant rank. We prove that under these assumptions, up to isometry of the ambient $G_2$ $7$-fold, this class consists of associative $3$-folds which are given by the quotients of either products of the form $\\Sigma\\times\\gamma$, where $\\Sigma$ is a complex curve in $X$ and $\\gamma$ is an appropriately chosen embedded circle in $T^3$, or by $3$-tori, $\\{x_0\\}\\times T^3$, where $x_0\\in X$.","abstract_html":"In this thesis, we study the possibility of extending the well-established construction of compact 7-manifolds carrying an In this thesis, we study the possibility of extending the well-established construction of compact $7$-manifolds carrying an irreducible torsion-free G2-structure, known as the \\emph{Twisted Connected Sum}, to the setting of $7$-orbifolds, spaces locally modeled on \\mathbb{R}7/\\Gamma, quotients of \\mathbb{R}7 by finite subgroups $\\Gamma$ of the group G2. Our work extends previous results by Alexei Kovalev (\\cite{Kov1}) and Dominic Joyce (\\cite{Joy1}) on the existence of torsion-free G2-structures and establishes a topological criterion for the irreducibility of such structures in the case of orbifolds. The strategy for the existence part is to lift the problem locally to a $\\Gamma$-invariant problem on a manifold. For irreducibility, the strategy is to adapt a criterion due to Joyce by considering a topological invariant for orbifolds called the \\emph{orbifold fundamental group}. We also investigate the irreducibility of a number of examples found in the literature, prove that the irreducibility of a global quotient of a G2-manifolds is equivalent to the irreducibility of the manifold, and construct a few dozen examples by using weighted projective spaces as the building blocks of the twisted connected sum. Another result in the thesis is a classification of associative $3$-folds in product G2-manifolds of the form X\\times T3, and related G2 orbifolds of the form (X\\times T3)/\\mathbb{Z}22 where $X$ is a hyper-Kähler $K3$ surface. The defining condition for this class is that the derivative of the torus projection has constant rank. We prove that under these assumptions, up to isometry of the ambient G2 $7$-fold, this class consists of associative $3$-folds which are given by the quotients of either products of the form \\Sigma\\timesγ, where $\\Sigma$ is a complex curve in $X$ and γ is an appropriately chosen embedded circle in T3, or by $3$-tori, \\{x0\\}\\times T3, where x0\\in X. Another result in the thesis is a classification of associative $3$-folds in product G2-manifolds of the form X\\times T3, and related G2 orbifolds of the form (X\\times T3)/\\mathbb{Z}22 where $X$ is a hyper-Kähler $K3$ surface. The defining condition for this class is that the derivative of the torus projection has constant rank. We prove that under these assumptions, up to isometry of the ambient G2 $7$-fold, this class consists of associative $3$-folds which are given by the quotients of either products of the form \\Sigma\\timesγ, where $\\Sigma$ is a complex curve in $X$ and γ is an appropriately chosen embedded circle in T3, or by $3$-tori, \\{x0\\}\\times T3, where x0\\in X.","abstract_has_math":true,"creators":["Barbuta, Andrei"],"institution":"University of Cambridge","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Kovalev, Alexei"],"committee_chairs":[],"committee_members":[],"year":2025,"date_issued":"2025-10-01","date_published":"2025-10-01","updated_at":"2026-07-22T22:24:30Z","subjects":["differential geometry","G2","Associative submanifolds","Twisted Connected Sum","Orbifolds"],"languages":["eng"],"rights":[],"rights_urls":["https://www.repository.cam.ac.uk/bitstreams/02cb62f5-805b-42cf-8673-e4a6e8e0ffc9/download","http://purl.org/NET/rdflicense/allrightsreserved"],"identifier_entries":[]},"links":{"outbound_url":"https://doi.org/10.17863/CAM.127855","outbound_label":"DOI","outbound_source":"dc:identifier.doi"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Kovalev, Alexei"]},{"key":"dc:creator","label":"Author","values":["Barbuta, Andrei"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2025-10-01"]},{"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/1810/399250"]},{"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":["differential geometry","G2","Associative submanifolds","Twisted Connected Sum","Orbifolds"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["https://www.repository.cam.ac.uk/bitstreams/02cb62f5-805b-42cf-8673-e4a6e8e0ffc9/download","http://purl.org/NET/rdflicense/allrightsreserved"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["https://doi.org/10.17863/CAM.127855"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://www.repository.cam.ac.uk/bitstreams/9e3f4275-8769-4147-bcdc-d91f8abcc25c/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["In this thesis, we study the possibility of extending the well-established construction of compact 7-manifolds carrying an In this thesis, we study the possibility of extending the well-established construction of compact $7$-manifolds carrying an irreducible torsion-free $G_2$-structure, known as the \\emph{Twisted Connected Sum}, to the setting of $7$-orbifolds, spaces locally modeled on $\\mathbb{R}^7/\\Gamma$, quotients of $\\mathbb{R}^7$ by finite subgroups $\\Gamma$ of the group $G_2$. Our work extends previous results by Alexei Kovalev (\\cite{Kov1}) and Dominic Joyce (\\cite{Joy1}) on the existence of torsion-free $G_2$-structures and establishes a topological criterion for the irreducibility of such structures in the case of orbifolds. The strategy for the existence part is to lift the problem locally to a $\\Gamma$-invariant problem on a manifold. For irreducibility, the strategy is to adapt a criterion due to Joyce by considering a topological invariant for orbifolds called the \\emph{orbifold fundamental group}. We also investigate the irreducibility of a number of examples found in the literature, prove that the irreducibility of a global quotient of a $G_2$-manifolds is equivalent to the irreducibility of the manifold, and construct a few dozen examples by using weighted projective spaces as the building blocks of the twisted connected sum. Another result in the thesis is a classification of associative $3$-folds in product $G_2$-manifolds of the form $X\\times T^3$, and related $G_2$ orbifolds of the form $(X\\times T^3)/\\mathbb{Z}_2^2$ where $X$ is a hyper-Kähler $K3$ surface. The defining condition for this class is that the derivative of the torus projection has constant rank. We prove that under these assumptions, up to isometry of the ambient $G_2$ $7$-fold, this class consists of associative $3$-folds which are given by the quotients of either products of the form $\\Sigma\\times\\gamma$, where $\\Sigma$ is a complex curve in $X$ and $\\gamma$ is an appropriately chosen embedded circle in $T^3$, or by $3$-tori, $\\{x_0\\}\\times T^3$, where $x_0\\in X$. Another result in the thesis is a classification of associative $3$-folds in product $G_2$-manifolds of the form $X\\times T^3$, and related $G_2$ orbifolds of the form $(X\\times T^3)/\\mathbb{Z}_2^2$ where $X$ is a hyper-Kähler $K3$ surface. The defining condition for this class is that the derivative of the torus projection has constant rank. We prove that under these assumptions, up to isometry of the ambient $G_2$ $7$-fold, this class consists of associative $3$-folds which are given by the quotients of either products of the form $\\Sigma\\times\\gamma$, where $\\Sigma$ is a complex curve in $X$ and $\\gamma$ is an appropriately chosen embedded circle in $T^3$, or by $3$-tori, $\\{x_0\\}\\times T^3$, where $x_0\\in X$."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["1b113bf8da206661664fb4d52a6cd449","87eda9de84448d1f82354d60eee3eb5f"]},{"key":"dc:title","label":"Title","values":["Compact G2-orbifolds via Twisted Connected Sums and Associative 3-folds"]}]}],"canonical_facts":{"dc:contributor.advisor":["Kovalev, Alexei"],"dc:creator":["Barbuta, Andrei"],"dc:date.issued":["2025-10-01"],"dc:description.abstract":["In this thesis, we study the possibility of extending the well-established construction of compact 7-manifolds carrying an In this thesis, we study the possibility of extending the well-established construction of compact $7$-manifolds carrying an irreducible torsion-free $G_2$-structure, known as the \\emph{Twisted Connected Sum}, to the setting of $7$-orbifolds, spaces locally modeled on $\\mathbb{R}^7/\\Gamma$, quotients of $\\mathbb{R}^7$ by finite subgroups $\\Gamma$ of the group $G_2$. Our work extends previous results by Alexei Kovalev (\\cite{Kov1}) and Dominic Joyce (\\cite{Joy1}) on the existence of torsion-free $G_2$-structures and establishes a topological criterion for the irreducibility of such structures in the case of orbifolds. The strategy for the existence part is to lift the problem locally to a $\\Gamma$-invariant problem on a manifold. For irreducibility, the strategy is to adapt a criterion due to Joyce by considering a topological invariant for orbifolds called the \\emph{orbifold fundamental group}. We also investigate the irreducibility of a number of examples found in the literature, prove that the irreducibility of a global quotient of a $G_2$-manifolds is equivalent to the irreducibility of the manifold, and construct a few dozen examples by using weighted projective spaces as the building blocks of the twisted connected sum. Another result in the thesis is a classification of associative $3$-folds in product $G_2$-manifolds of the form $X\\times T^3$, and related $G_2$ orbifolds of the form $(X\\times T^3)/\\mathbb{Z}_2^2$ where $X$ is a hyper-Kähler $K3$ surface. The defining condition for this class is that the derivative of the torus projection has constant rank. We prove that under these assumptions, up to isometry of the ambient $G_2$ $7$-fold, this class consists of associative $3$-folds which are given by the quotients of either products of the form $\\Sigma\\times\\gamma$, where $\\Sigma$ is a complex curve in $X$ and $\\gamma$ is an appropriately chosen embedded circle in $T^3$, or by $3$-tori, $\\{x_0\\}\\times T^3$, where $x_0\\in X$. Another result in the thesis is a classification of associative $3$-folds in product $G_2$-manifolds of the form $X\\times T^3$, and related $G_2$ orbifolds of the form $(X\\times T^3)/\\mathbb{Z}_2^2$ where $X$ is a hyper-Kähler $K3$ surface. The defining condition for this class is that the derivative of the torus projection has constant rank. We prove that under these assumptions, up to isometry of the ambient $G_2$ $7$-fold, this class consists of associative $3$-folds which are given by the quotients of either products of the form $\\Sigma\\times\\gamma$, where $\\Sigma$ is a complex curve in $X$ and $\\gamma$ is an appropriately chosen embedded circle in $T^3$, or by $3$-tori, $\\{x_0\\}\\times T^3$, where $x_0\\in X$."],"dc:format.checksum.md5":["1b113bf8da206661664fb4d52a6cd449","87eda9de84448d1f82354d60eee3eb5f"],"dc:identifier.doi":["https://doi.org/10.17863/CAM.127855"],"dc:identifier.uri":["https://www.repository.cam.ac.uk/bitstreams/9e3f4275-8769-4147-bcdc-d91f8abcc25c/download"],"dc:language":["eng"],"dc:publisher.institution":["University of Cambridge"],"dc:relation.isreferencedby.uri":["https://www.repository.cam.ac.uk/handle/1810/399250"],"dc:rights":["https://www.repository.cam.ac.uk/bitstreams/02cb62f5-805b-42cf-8673-e4a6e8e0ffc9/download","http://purl.org/NET/rdflicense/allrightsreserved"],"dc:subject":["differential geometry","G2","Associative submanifolds","Twisted Connected Sum","Orbifolds"],"dc:title":["Compact G2-orbifolds via Twisted Connected Sums and Associative 3-folds"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["Doctoral"],"dc:type.qualificationname":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-22T22:24:30Z"}