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Diarmuid Crowley: Lectures

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 Nordström and
Sebastian Goette.