Rotation index, Milnor--Munkres--Novikov pairing, and group actions on manifolds

We introduce an invariant of a pair of commuting invertible matrices that we call the rotation index. We apply this invariant, together with the Milnor–Munkres–Novikov pairing, to the study of some questions about group actions of ${\mathbb{Z}}^2$, specifically the Nielsen realization problem, higher-rank Anosov actions, and extending actions from the sphere $S^{d-1}$ to the disk $D^d$.