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

##
Lucía Martín Merchán

created by daniele on 05 Oct 2021

modified on 13 Nov 2021

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 G_{2} 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 G_{2} 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 G_{2} structure that cannot be endowed with any torsion-free G_{2} 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 G_{2} structure that admits an involution preserving φ such that the quotient M/Z_{2} is a non-formal orbifold with b_{1=1.} Then we perform a resolution of these singularities obtaining a manifold endowed with a closed G_{2} structure; we finally prove that the resolution verifies the same topological properties and do not admit any torsion-free G_{2} structure.