An equivariant analog of the Poincaré–Hopf theorem

A new method for localization of algebro-topological invariants of smooth manifolds is given in terms of equivariant tangent vector fields. Main realizations of direct image constructions – the Gysin map and the Becker–Gottlieb transfer map – are calculated for Grassmannizations of complex vector bundles and for a complex-oriented cohomology theory.