The index of an equivariant vector field and addition theorems for Pontryagin classes

In this paper we construct a theory of indices of Morse–Bott vector fields on a manifold and use it to solve a famous localization problem for the transfer map. As a consequence, we obtain addition theorems for universal Pontryagin classes in cobordisms. This enables us to complete the construction of the theory of universal characteristic classes, which was begun more than twenty years ago.