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

2019/05/31 by Harring, Paula, Köhl, Ralf
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1905.13444

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 π₁(G) = π₁(K). It thus suffices to determine π₁(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 π₁(K) more generally also holds in the non-symmetrizable two-spherical situation.

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