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We determine the fundamental groups of symmetrizable algebraically simplyconnected 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 weachieve by investigating the fundamental groups of generalized flag varieties.Our results apply in all cases in which the Bruhat decomposition of thegeneralized flag variety is a CW decomposition $-$ in particular, we cover thecomplete symmetrizable situation; the result concerning the structure of$\pi_1(K)$ more generally also holds in the non-symmetrizable two-sphericalsituation.

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