Fold cobordisms and a Poincaré–Hopf-type theorem for the signature

We give complete geometric invariants of cobordisms framed fold maps. These consist two types. take the immersion singular set into target manifold together with information about non-triviality normal bundle in source manifold. were introduced author's earlier works. Secondly we induced stable partial framing on whose studied general by Koschorke. show that these describe completely cobordism groups maps R^n. Then are looking for dependencies between and study 4k-dimensional manifolds R^2. construct special which representatives classes also compute groups. obtain a Poincare-Hopf type formula, connects local data singularities map an oriented M to signature M. unoriented case analogously prove similar formula twisting set.