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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Acknowledgements
We thank Jean-Michel Bismut, Uli Bunke, Xianzhe Dai, Matthias Lesch for inspiring discussions on adiabatic limits and variational formulas for \(\eta \)-invariants on manifolds with boundary. We thank Alessio Corti, Jesus Martinez Garcia, David Morrison, Emanuel Scheidegger and Katrin Wendland for helpful information about K3-surfaces and Fano threefolds. We also thank Mark Haskins and Arkadi Schelling for talking with us about \(G_2\)-manifolds and \(G_2\)-bordism. We are particularly indebted to Don Zagier for his formula for \(F_{k,\varepsilon }(s)\) in Sect. 4.3. SG and JN would like to thank the Simons foundation for its support of their research under the Simons Collaboration on “Special Holonomy in Geometry, Analysis and Physics” (grants #488617, Sebastian Goette, and #488631, Johannes Nordström).
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Crowley, D., Goette, S., Nordström, J. (2020). Distinguishing \(G_2\)-Manifolds. In: Karigiannis, S., Leung, N., Lotay, J. (eds) Lectures and Surveys on G2-Manifolds and Related Topics. Fields Institute Communications, vol 84. Springer, New York, NY. https://doi.org/10.1007/978-1-0716-0577-6_6
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