Geometria Complessa e Geometria Differenziale
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A compact non-formal closed $G_2$ manifold with $b_1=1$

Lucía Martín Merchán

created by bazzoni on 14 Jan 2022
modified on 25 Jan 2022

26 jan 2022 -- 17:00

Geometry in Como - online

Abstract.

A $G_2$ structure on a 7-dimensional Riemannian manifold determined by a certain type of 3-form $\varphi$. These are classified into 16 types according to PDEs involving $\varphi$; for instance, the $G_2$ structure is torsion-free if $\varphi$ is parallel, closed if $\varphi$ is closed and cocalibrated if $\varphi$ 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 $(M,\varphi)$ with a closed $G_2$ structure that admits an involution preserving $\varphi$ such that the quotient $M/\mathbb{Z}_2$ 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.

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