Algebraic topology of $G_2$ manifolds

TL;DR

This paper surveys the topology of Grassmannian bundles related to the exceptional Lie group G_2, including new results, with applications to the existence of special submanifolds in G_2 manifolds relevant to calibrated geometry.

Contribution

It provides a self-contained survey of the topology of G_2-related Grassmannian bundles, including new results and proofs, facilitating future research in calibrated geometry and G_2 dualities.

Findings

New topological results on Grassmannian bundles related to G_2

Existence proofs for special submanifolds in G_2 manifolds

Self-contained proofs for foundational results

Abstract

In this paper we give a survey of various results about the topology of oriented Grassmannian bundles related to the exceptional Lie group G_2. Some of these results are new. We give self-contained proofs here. One often encounters these spaces when studying submanifolds of manifolds with calibrated geometries. For the sake of completeness we decided to collect them here in a self-contained way to be easily accessible for future usage in calibrated geometry. As an application we deduce existence of certain special 3 and 4 dimensional submanifolds of G_2 manifolds with special properties, which appear in the first named author's work with S. Salur about G_2 dualities.