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This paper uses algebro-topological techniques such as characteristic classesand obstruction theory, together with the $h$-principles for$\widetilde{\mathrm{G}}_2$ and $\mathrm{SL}(3;\mathbb{R})^2$ forms recentlyestablished by the author and the $h$-principle for $\mathrm{SL}(3;\mathbb{C})$forms established by Donaldson, to prove results on the topological propertiesof 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 aclosed (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 isintroduced, 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 analoguein the $\mathrm{G}_2$ case; and a complete classification of closed$\mathrm{SL}(3;\mathbb{C})$ forms up to homotopy is provided. Additionally, alower bound on the number of homotopy classes of closed$\mathrm{SL}(3;\mathbb{R})^2$ forms on a given manifold is obtained, and thequestion 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.

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