Compact G₂-orbifolds via Twisted Connected Sums and Associative 3-folds

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.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2025

Author dc:creator

• Barbuta, Andrei

Advisor dc:contributor.advisor

• Kovalev, Alexei

• differential geometry
• G₂
• Associative submanifolds
• Twisted Connected Sum
• Orbifolds

Licence

• https://www.repository.cam.ac.uk/bitstreams/02cb62f5-805b-42cf-8673-e4a6e8e0ffc9/download
• http://purl.org/NET/rdflicense/allrightsreserved

Language dc:language

eng

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.127855

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/399250