Gauge Theory and Topology Seminar: Nearly-parallel $G_2$-structures in different homotopy classes

A $G_2$-structure on a closed 7-manifold is a 3-form of a special type pointwise modelled on the cross product of the imaginary octonions, and the space of such forms can be disconnected. Crowley and Nordström introduced invariants $\nu$ and $\xi$, defined via characteristic numbers of a spin coboundary, that detect its components; while their main goal was the case of parallel $G_2$-structures, we are concerned with the nearly-parallel case. A natural test case is the family of Aloff–Wallach spaces $N_{k,l} = SU(3)/S^1_{k,l}$, which carry nearly-parallel $G_2$-structures and among which different spaces can be diffeomorphic. Using an intrinsic formula for $\xi$ together with Goette’s techniques for homogeneous spaces, we show $\nu(\varphi) = 0$ and $\xi(\varphi) = \frac{3}{2}kl(k+l)$ on $N_{k,l}$. These computations give explicit closed 7-manifolds carrying two nearly-parallel $G_2$-structures, that are not homotopic.