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 $n \in \mathbb{N}$?



Homotopy groups of Fredholm operators
EDIT (much later): My answer below is not quite correct, as pointed out by Dan Ramras, because I ignored the importance of base points in the definition of $\pi\_n$. See Ramras' answer for the needed corrections.




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It is a theorem (due I think to Atiyah) that $\mathcal{F}$ is the classifying space for the topological K-theory functor:
$$[X,\mathcal{F}] \cong K(X)$$
for any space $X$. The isomorphism is given as follows: given a map $T \colon X \to \mathcal{F}$, deform $T$ so that $\dim \ker(T(x))$ is constant and assign $T$ to the K-theory class $[\ker T] - [\text{coker}\, T]$. It follows that $\pi\_n(\mathcal{F}) \cong K(S^n)$ which can be calculated using Bott periodicity.


