Topological properties of closed $\widetilde{\mathrm{G}}_2$, $\mathrm{SL}(3;\mathbb{C})$ and $\mathrm{SL}(3;\mathbb{R})^2$ forms on manifolds

TL;DR

This paper investigates the topological characteristics of special geometric structures on 6- and 7-manifolds, establishing criteria for their existence, classifying forms, and exploring cobordism relations using advanced topological techniques.

Contribution

It introduces new topological criteria and classifications for closed $ ilde{G}_2$, $ ext{SL}(3; ext{C})$, and $ ext{SL}(3; ext{R})^2$ forms, extending previous results and conjectures in the field.

Findings

Criterion for 7-manifolds to admit closed $ ilde{G}_2$-structures.

Complete classification of closed $ ext{SL}(3; ext{C})$ forms up to homotopy.

Homotopic $ ext{SL}(3; ext{C})$ and $ ext{SL}(3; ext{R})^2$ forms are $ ilde{G}_2$-cobordant.

Abstract

This paper uses algebro-topological techniques such as characteristic classes and obstruction theory, together with the $h$-principles for $\widetilde{\mathrm{G}}_2$ and $\mathrm{SL}(3;\mathbb{R})^2$ forms recently established by the author and the $h$-principle for $\mathrm{SL}(3;\mathbb{C})$ forms established by Donaldson, to prove results on the topological properties of closed $\widetilde{\mathrm{G}}_2$, $\mathrm{SL}(3;\mathbb{C})$ and $\mathrm{SL}(3;\mathbb{R})^2$ forms on oriented 6- and 7-manifolds. Specifically, a criterion for an arbitrary oriented 7-manifold to admit a closed (resp. coclosed) $\widetilde{\mathrm{G}}_2$-structure is obtained, proving a conjecture of L\^{e}; a generalisation of Donaldson's '$\mathrm{G}_2$-cobordisms' to $\widetilde{\mathrm{G}}_2$, $\mathrm{SL}(3;\mathbb{C})$ and $\mathrm{SL}(3;\mathbb{R})^2$ forms is introduced, with homotopic $\mathrm{SL}(3;\mathbb{C})$ and $\mathrm{SL}(3;\mathbb{R})^2$ forms in a given cohomology class shown to be $\widetilde{\mathrm{G}}_2$-cobordant, a result which currently has no analogue in the $\mathrm{G}_2$ case; and a complete classification of closed $\mathrm{SL}(3;\mathbb{C})$ forms up to homotopy is provided. Additionally, a lower bound on the number of homotopy classes of closed $\mathrm{SL}(3;\mathbb{R})^2$ forms on a given manifold is obtained, and the question of which closed $\mathrm{SL}(3;\mathbb{C})$ or $\mathrm{SL}(3;\mathbb{R})^2$ forms arise as the boundary values of closed $\widetilde{\mathrm{G}}_2$-structures on oriented 7-manifolds is investigated.