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

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.

Mathematics

Audience: researchers in the topic

Geometry Seminar - University of Florence

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