  We consider the space $\X$ of Anosov diffeomorphisms homotopic to a fixed
automorphism $L$ of an infranilmanifold $M$. We show that if $M$ is the 2-torus
$\mathbb T^2$ then $\X$ is homotopy equivalent to $\mathbb T^2$. In contrast,
if dimension of $M$ is large enough, we show that $\X$ is rich in homotopy and
has infinitely many connected components.
