### 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 (i.e., -manifolds). We would like to use this wealth of examples to guess further properties of -manifolds and to find obstructions against holonomy , taking into account the underlying topological -structures.

The Crowley-Nordström v-invariant distinguishes topological -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 -bordism presents no obstruction against holonomy . We also exhibit examples of 7-manifolds with disconnected -moduli space. Our computation of the v-invariants involves integration of the Bismuth-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.

### September 12,2017

TITLE: The extended *v* -invariant — progress and problems

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

### June 7, 2017

TITLE: Disconnecting the G_{2} moduli space

### September 6, 2016

TITLE: Connected components of the moduli space of G_{2} manifolds

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

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