# Character on a homogeneous space

## TL;DR

This paper investigates the cohomological triviality of fibrations in homogeneous spaces of affine algebraic groups over complex numbers, using topological methods, and extends results to broader classes of algebraic groups, providing structural insights.

## Contribution

It introduces a topological approach to analyze cohomological triviality in homogeneous spaces and generalizes results to arbitrary connected algebraic groups, including a structure theorem for quasi-reductive groups.

## Key findings

- Cohomological triviality characterized via covering space theory.
- Generalization of results to all connected algebraic groups.
- Structural classification of quasi-reductive algebraic groups.

## Abstract

In this paper we look at the notion of cohomological triviality of fibrations of homogeneous spaces of affine algebraic groups defined over $\mathbb{C}$ and use topological methods, primarily the theory of covering spaces. This is made possible because of the structure theory of affine algebraic groups. Further, we generalize our results for arbitrary connected algebraic groups and their homogeneous spaces. As an application of our methods, we give a structure result for quasi-reductive algebraic groups(i.e groups whose unipotent radical is trivial), upto isogeny.