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Sebastian Goette: Lectures

September 12,2017
TITLE: The extended v -invariant — progress and problems

ABSTRACT: We recall the definition of the extended \nu-invariant, which distinguishes connected components of the moduli space of compact Riemannian manifolds whose holonomy group is G_2 (G_2-manifolds for short). All known computations of \nu(M,g) for G_2-manifolds (M,g) give values that are divisible by three. This implies that all these examples are topologically G_2-nullbordant. It is therefore interesting to know if 3|\nu(M,g) for all G_2-manifolds. We will report on results and projects related to this question.

June 7, 2017
TITLE: Disconnecting the G2 moduli space

September 6, 2016
TITLE: Connected components of the moduli space of G2 manifolds

ABSTRACT: The Crowley-Nordström \nu-invariant distinguishes topological G_2 structures on 7-manifolds. It takes values in \mathbb Z /48. There is a \mathbb Z-valued extension for manifolds of holonomy G_2. We will introduce both invariants and show how they can be computed for extra twisted connected sums using \nu-invariants of Dirac operators. This allows us to exhibit examples of 7-manifolds M where the space of G_2-holonomy metrics is disconnected.

We will then talk about some related open questions and problems and sketch possible next steps in our research program.