Higher homotopy groups of topological Kac-Moody groups, Whitehead tower, and string groups

This note establishes that homotopy groups of topological split real Kac-Moody groups are countable and, hence, concludes the existence of Whitehead towers consisting of topological groups for these groups and their maximal compact subgroups. Moreover, this note proposes a construction for string groups of the $E_n$-series.