### June 10, 2022

TITLE: A prolegomena to finding higher homotopy in G_2-moduli spaces

ABSTRACT: I will begin this talk by reviewing topological approach to studying

the connected components

of G_2-moduli spaces via the \nu-invariant. Then I will look at the

possibility of detecting higher homotopy

groups in G_2-moduli spaces.

The idea is to define higher-dimensional \nu-invariants, which can detect

homotopy classes in \pi_i(G_2(M)) (where G_2(M) is the space of G_2-structures

on M) and even on the quotient \pi_i(G_2(M)) / \pi_i(\Diff(M)).

At the time of writing, passing to \pi_i of the actual moduli space G_2(M) /

Diff(M) or finding essential maps to the moduli space of torsion free

G_2-structures are open problems but we hope the methods may none the less be

of interest.

I will also discuss related results on the mapping class group of M and some

higher homotopy groups of Diff(M).

This is an early report on work in progress with Johannes NordstrÃ¶m and

Sebastian Goette.