A compact non-formal closed G_2 manifold with b_1=1

Lucía Martín Merchán

23 nov 2021 -- 14:30

DIMaI, Firenze (online)

Seminario di Geometria Differenziale e Analisi Complessa del Dipartimento di Matematica e Informatica "Ulisse Dini" dell'Università di Firenze

Abstract.

A G₂ structure on a 7-dimensional Riemannian manifold determined by a certain type of 3-form φ. These are classified into 16 types according to PDEs involving φ; for instance, the G₂ structure is torsion-free if φ is parallel, closed if φ is closed and cocalibrated if φ is co-closed. This talk contributes to understanding topological properties of compact manifolds with a closed G₂ structure that cannot be endowed with any torsion-free G₂ structure. Namely, we construct such a manifold that is non-formal and has first Betti number b_1=1. The starting point is a nilmanifold (M,φ) with a closed G₂ structure that admits an involution preserving φ such that the quotient M/Z₂ is a non-formal orbifold with b_1=1. Then we perform a resolution of these singularities obtaining a manifold endowed with a closed G₂ structure; we finally prove that the resolution verifies the same topological properties and do not admit any torsion-free G₂ structure.