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On the \texorpdfstring{$\nu$}{nu}-invariant of two-step nilmanifolds with closed \texorpdfstring{$\mathrm G_2$}{G2}-structure
For every non-vanishing spinor field on a Riemannian $7$-manifold, Crowley, Goette, and Nordstr\"om introduced the so-called $\nu$-invariant. This is an integer modulo $48$, and can be defined in terms of Mathai--Quillen currents, harmonic spinors, and $\eta$-invariants of spin Dirac and odd-signature operator. We compute these data for the compact two-step …