  $Hom(G,H)$ is a polyhedral complex defined for any two undirected graphs $G$
and $H$. This construction was introduced by Lov\'asz to give lower bounds for
chromatic numbers of graphs. In this paper we initiate the study of the
topological properties of this class of complexes. We prove that $Hom(K_m,K_n)$
is homotopy equivalent to a wedge of $(n-m)$-dimensional spheres, and provide
an enumeration formula for the number of the spheres. As a corollary we prove
that if for some graph $G$, and integers $m\geq 2$ and $k\geq -1$, we have
$\varpi_1^k(\thom(K_m,G))\neq 0$, then $\chi(G)\geq k+m$; here $Z_2$-action is
induced by the swapping of two vertices in $K_m$, and $\varpi_1$ is the first
Stiefel-Whitney class corresponding to this action. Furthermore, we prove that
a fold in the first argument of $Hom(G,H)$ induces a homotopy equivalence. It
then follows that $Hom(F,K_n)$ is homotopy equivalent to a direct product of
$(n-2)$-dimensional spheres, while $Hom(\bar{F},K_n)$ is homotopy equivalent to
a wedge of spheres, where $F$ is an arbitrary forest and $\bar{F}$ is its
complement.
