The space of Anosov diffeomorphisms

We consider the space $\text{X}$ 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 $\text{X}$ is homotopy equivalent to ${\mathbb{T}}^2$. In contrast, if dimension of $M$ is large enough, we show that $\text{X}$ is rich in homotopy and has infinitely many connected components.