  We study the topology of spaces related to Kac-Moody groups. Given a split
Kac-Moody group over the complex numbers, let K denote the unitary form with
maximal torus T having normalizer N(T). In this article we study the cohomology
of the flag manifold K/T, as a module over the Nil-Hecke ring, as well as the
(co)homology of K as a Hopf algebra. In particular, if F is a field of positive
characteristic, we show that H_*(K,F) is a finitely generated algebra, and that
H^*(K,F) is finitely generated only if K is a compact Lie group . We also study
the stable homotopy type of the classifying space BK and show that it is a
retract of the classifying space BN(T). We illustrate our results with the
example of rank two Kac-Moody groups.
