If $X$ is separable complex Hilbert space and $\mathcal{F}$ the topological space of Fredholm operators on $X$, then it is well-known, that $$ \pi_0(\mathcal{F}) = \mathbb{Z}\, , $$ i.e. the connected components are classified by the index of the Fredholm operator.

But what is about higher homotopy groups? What is known about $\pi_n(\mathcal{F})$ for $n \in \mathbb{N}$?