  This paper extends certain characterizations of nullhomotopic maps between
p-compact groups to maps with target the p-completed classifying space of a
connected Kac-Moody group and source the classifying space of either a
p-compact group or a connected Kac-Moody group. A well known inductive
principle for p-compact groups is applied to obtain general, mapping space
level results. An arithmetic fiber square computation shows that a null map
from the classifying space of a connected compact Lie group to the classifying
space of a connected topological Kac-Moody group can be detected by restricting
to the maximal torus. Null maps between the classifying spaces of connected
topological Kac-Moody groups cannot, in general, be detected by restricting to
the maximal torus due to the nonvanishing of an explicit abelian group of
obstructions described here. Nevertheless, partial results are obtained via the
application of algebraic discrete Morse theory to higher derived limit
calculations. These partial results show that null maps are detected by
restricting to the maximal torus in many cases of interest.
