# Fundamental groups of split real Kac-Moody groups and generalized real flag manifolds. With appendices by Tobias Hartnick and Ralf K\"ohl and by Julius Gr\"uning and Ralf K\"ohl

## TL;DR

This paper computes the fundamental groups of split real Kac-Moody groups with the Kac-Peterson topology by analyzing their flag varieties, extending known results from finite-dimensional cases to infinite-dimensional Kac-Moody groups.

## Contribution

It determines the fundamental groups of symmetrizable split real Kac-Moody groups and their flag varieties, generalizing finite-dimensional results to infinite-dimensional Kac-Moody settings.

## Key findings

- Fundamental groups of Kac-Moody groups are determined via flag varieties.
- Results apply to all symmetrizable cases with CW decompositions.
- The structure of fundamental groups extends to non-symmetrizable two-spherical cases.

## Abstract

We determine the fundamental groups of symmetrizable algebraically simply connected split real Kac-Moody groups endowed with the Kac-Peterson topology. In analogy to the finite-dimensional situation, the Iwasawa decomposition $G = KAU_+$ provides a weak homotopy equivalence $K \hookrightarrow G$, implying $\pi_1(G) = \pi_1(K)$. It thus suffices to determine $\pi_1(K)$ which we achieve by investigating the fundamental groups of generalized flag varieties. Our results apply in all cases in which the Bruhat decomposition of the generalized flag variety is a CW decomposition $-$ in particular, we cover the complete symmetrizable situation; the result concerning the structure of $\pi_1(K)$ more generally also holds in the non-symmetrizable two-spherical situation.