Computing $\nu$-invariants of Joyce's compact $G_2$-manifolds

TL;DR

This paper computes the $ u$-invariant for many Joyce's constructed closed torsion-free $G_2$-manifolds, advancing understanding of their topological invariants.

Contribution

It provides explicit calculations of the $ u$-invariant for a broad class of Joyce's $G_2$-manifolds using spectral methods.

Findings

Computed $ u$-invariants for numerous Joyce $G_2$-manifolds

Demonstrated the applicability of spectral descriptions to topological invariants

Enhanced understanding of the topology of $G_2$-manifolds

Abstract

Crowley and Nordstr\"{o}m introduced an invariant of $G_2$-structures on the tangent bundle of a closed 7-manifold, taking values in the integers modulo 48. Using the spectral description of this invariant due to Crowley, Goette and Nordstr\"{o}m, we compute it for many of the closed torsion-free $G_2$-manifolds defined by Joyce's generalized Kummer construction.