Fundamental groups of split real Kac-Moody groups and generalized real flag manifolds. With appendices by Tobias Hartnick and Ralf Köhl and by Julius Grüning and Ralf Köhl
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 $π_1(G) = π_1(K)$. It thus suffices to determine $π_1(K)$ 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 $π_1(K)$ more generally also holds in the non-symmetrizable two-spherical situation.