Topological properties of closed $\mathrm{G}_2$ manifolds through compact quotients of Lie groups

Abstract: A $\mathrm{G}_2$ structure on a 7-dimensional Riemannian manifold $(M,g)$ is determined by a stable of 3-form $\varphi$. It is said to be closed if $d\varphi=0$ and torsion-free if $\varphi$ is parallel. The purpose of this talk is to discuss two problems where compact quotients of Lie groups are useful for understanding topological properties of compact closed $\mathrm{G}_2$ manifolds that don´t admit any torsion-free $\mathrm{G}_2$ structure. More precisely, these problems are related to the open questions: Are simply connected compact closed $\mathrm{G}_2$ manifolds almost formal? Could a compact closed $\mathrm{G}_2$ manifold have third Betti number $b_3=0$?

Using compact quotients of Lie groups, we first outline the construction of a manifold admitting a closed $\mathrm{G}_2$ structure that is not almost formal and has first Betti number $b_1=1$. Later, we show that there aren´t invariant exact $\mathrm{G}_2$ structures on compact quotients of Lie groups. The last result is joint work with Anna Fino and Alberto Raffero.

differential geometrygeometric topologymetric geometry

Audience: researchers in the topic

( video )

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Virtual seminar on geometry with symmetries

Series comments: Description: Research seminar in Lie group actions in Riemannian geometry.

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