Distinguishing G_2-manifolds
Diarmuid Crowley, Sebastian Goette, Johannes Nordstr\"om
TL;DR
This survey discusses invariants and construction methods to differentiate connected components and homeomorphic but non-diffeomorphic G_2-manifolds on closed 7-manifolds.
Contribution
It provides a comprehensive overview of invariants and construction techniques for distinguishing G_2-manifolds, including twisted connected sum methods.
Findings
Invariants can distinguish different G_2-structures.
Construction methods realize G_2-manifolds with distinct invariants.
The survey summarizes current techniques and open problems.
Abstract
In this survey, we describe invariants that can be used to distinguish connected components of the moduli space of holonomy G_2 metrics on a closed 7-manifold, or to distinguish G_2-manifolds that are homeomorphic but not diffeomorphic. We also describe the twisted connected sum and extra-twisted connected sum constructions used to realise G_2-manifolds for which the above invariants differ.
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Taxonomy
TopicsGeometry and complex manifolds · Geometric Analysis and Curvature Flows · Geometric and Algebraic Topology