# KO-theory of complex flag varieties of ordinary type

## TL;DR

This paper computes the topological Witt groups and KO-groups of complex flag varieties of ordinary type, providing detailed algebraic structures for types A, B, and C, using an approach by Zibrowius.

## Contribution

It offers the first comprehensive computation of topological Witt groups for all complex flag varieties of ordinary type, including their multiplicative structures.

## Key findings

- Computed topological Witt groups for all complex flag varieties of ordinary type.
- Provided explicit descriptions of the graded Witt rings for types A, B, and C.
- Connected the computations to Balmer's Witt groups over algebraically closed fields.

## Abstract

We compute the topological Witt groups of every complex flag manifold of ordinary type, and thus the interesting (i.e. torsion) part of the KO-groups of these manifolds. Equivalently, we compute Balmer's Witt groups of each flag variety of ordinary type over an algebraically closed field of characteristic not two. Our computation is based on an approach developed by Zibrowius. For types A, B and C, we obtain a full description not only of the additive but also of the multiplicative structure of the graded Witt rings.