Characteristic classes as complete obstructions

TL;DR

This paper offers a unified interpretation of characteristic classes as obstructions to reducing structure groups and extends classical theorems, introducing invariants that quantify possible group reductions of principal bundles.

Contribution

It provides a novel, uniform framework for understanding characteristic classes as obstructions and introduces invariants that measure the number of group reductions.

Findings

Characteristic classes are interpreted as obstructions to structure group reduction.

Introduces invariants that count the number of principal bundle reductions.

Establishes a long exact sequence linking these invariants with cohomology groups.

Abstract

In the first part of this paper, we propose a uniform interpretation of characteristic classes as obstructions to the reduction of the structure group and to the existence of an equivariant extension of a certain homomorphism defined a priori only on a single fiber of the bundle. By plugging in the correct parameters, we recover several classical theorems. Afterwards, we define a family of invariants of principal bundles that detect the number of group reductions that a principal bundle admits. We prove that they fit into a long exact sequence of abelian groups, together with the cohomology of the base space and the cohomology of the classifying space of the structure group.