  We give a counterexample to a conjecture of D.H. Gottlieb and prove a
strengthened version of it. The conjecture says that a map from a finite
CW-complex X to an aspherical CW-complex Y with non-zero Euler characteristic
can have non-trivial degree (suitably defined) only if the centralizer of the
image of the fundamental group of X is trivial. As a corollary we show that in
the above situation all components of non-zero degree maps in the space of maps
from X to Y are contractible.
