It is sometimes possible
to prove that a functor is homotopy invariant using only a knowledge of
the domain and range categories of the functor. It is known, for example,
that every covariant or contravariant functor from the category of simplicial
complexes (with continuous mappings) to the category of countable groups is
homotopy invariant. This result has been extended to covariant, but not
contravariant, functors with domain the category of smooth manifolds. In
the contravariant case, the proof breaks down because certain mappings
are not differentiable. This fault will be corrected in this paper. Among
other results, it will be shown that every contravariant functor from the
category of smooth manifolds to the category of countable groups is homotopy
invariant.
