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Sebastian Goette: Lectures

September 7, 2023
TITLE: Homotopy Associative Submanifolds in G2-manifolds

ABSTRACT: Associative submanifolds are certain calibrated submanifolds in G2-manifolds. There is the hope that counting them will reveal subtle information about the underlying G2-structure. On the other hand, certain singular associatives can be resolved in exactly three different ways, so a naive count will be meaningless. In this talk, we will define homotopy associatives as cobordism classes of three-dimensional submanifolds that are adapted to the G2-structure in a rather weak sense. We will see that a given cobordism class can be interpreted as a homotopy associative in exactly three different ways. This might help us to define a consistent counting scheme even when the naive number of associatives in a given cobordism class changes due to singularities.

Slides of Lecture

September 11, 2022
TITLE: Some examples of extra twisted connected sum G_2-manifolds

ABSTRACT: The twisted connected sum construction is one of a few known ways to produce compact G_2-manifolds. Extra twisted connected sums form a slight generalisation. They have been used to show that the moduli space of G_2-metrics on a given 7-manifold can be disconnected, even if one fixes a connected component of the space of topological G_2-structures. In this talk, I want to present some more details of the construction. In particular, I want to present quotients and coverings of extra twisted connected sums, as well as a kind of t-duality.

Slides of Lecture

April 11, 2019
TITLE: Distinguishing G2-manifolds

ABSTRACT: There are several invariants from differential topology that
distinguish smooth 7-manifolds and G2-structures on them. I will give a
short introduction, then focus on the nu invariant and its analytic

September 13, 2018
TITLE: Extra twisted connected sums and their v -invariants

ABSTRACT: Joyce’s orbifold construction and the twisted connected sums by Kovalev and Corti-Haskins-Nordström-Pacini provide many examples of compact Riemannian 7-manifolds with holonomy G_2 (i.e., G_2-manifolds). We would like to use this wealth of examples to guess further properties of G_2-manifolds and to find obstructions against holonomy G_2, taking into account the underlying topological G_2-structures.

The Crowley-Nordström v-invariant distinguishes topological G_2-structures. It vanishes for all twisted connected sums. By adding an extra twist to this construction, we show that the v-invariant can assume all of its 48 possible values. This shows that G_2-bordism presents no obstruction against holonomy G_2. We also exhibit examples of 7-manifolds with disconnected G_2-moduli space. Our computation of the v-invariants involves integration of the Bismut-Cheeger η-forms for families of tori, which can be done either by elementary hyperbolic geometry, or using modular properties of the Dedekind η-function.

Slides of lecture

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.