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 $x_0\in X$.