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.