# Regularity theorem for totally nonnegative flag varieties

## TL;DR

This paper proves that the totally nonnegative parts of partial flag varieties are regular CW complexes, confirming conjectures about their topological structure and the homeomorphism of positroid cell closures to balls.

## Contribution

It establishes the regular CW complex structure of totally nonnegative flag varieties, confirming conjectures by Williams and Postnikov about their topology.

## Key findings

- Totally nonnegative flag varieties are regular CW complexes.
- Closure of each positroid cell is homeomorphic to a ball.
- Confirms conjectures of Williams and Postnikov.

## Abstract

We show that the totally nonnegative part of a partial flag variety $G/P$ (in the sense of Lusztig) is a regular CW complex, confirming a conjecture of Williams. In particular, the closure of each positroid cell inside the totally nonnegative Grassmannian is homeomorphic to a ball, confirming a conjecture of Postnikov.