An Index of an Equivariant Vector Field and Addition Theorems for Pontrjagin Characteristic Classes

TL;DR

This paper develops an index theory for Morse--Bott vector fields, solves a localization problem for transfer maps, and derives addition theorems for Pontrjagin classes, advancing the universal characteristic classes' theory.

Contribution

It introduces an index of equivariant vector fields and proves addition theorems for Pontrjagin classes, completing a long-standing construction in characteristic classes.

Findings

Constructed index theory for Morse--Bott vector fields

Solved localization problem for transfer maps

Derived addition theorems for Pontrjagin classes

Abstract

The theory of indices of Morse--Bott vector fields on a manifold is constructed and the famous localization problem for the transfer map is solved on its base in the present paper. As a consequence, we obtained addition theorems for the universal Pontrjagin characteristic classes in cobordisms. These results gave us a possibility to complete the construction, which was begun more than twenty years ago, of the universal characteristic classes' theory.