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The size of some vanishing and critical sets

We prove that the vanishing sets of all top forms on a non-orientable manifold are at least  1-dimensional in the general case and at most 1-codimensional in the compact case. We apply these facts to show that the critical sets of some differentiable maps are at  least 1-dimensional in the general case and at most 1-codimensional when the source manifold is compact.