TITLE: `Naturally occuring'  $K(\pi, n)$ spaces, for $n \geq 2$.
QUESTION [42 upvotes]: [edited!] Given a group $\pi$ and an integer $n>1$, what are examples of Eilenberg-Maclane spaces $K(\pi, n)$ that can be constructed as "known" manifolds? (or if not a manifold, say some space people had a pre-existing desire to study before $K(\pi,n)$ spaces were identified as being of interest)

Constructing $K({\bf Z}, 2)$ as ${\bf CP}^{\infty}$ is the only example I know - but there must be more out there.
I'm interested in concrete examples (like the one above) that could, e.g., be given in a Topics grad course for topology students. They seem to be scarse, so it would be nice to know what was known.
Note: I've excluded $n=1$ because most people know examples (or can figure them out) in this case.

REPLY [32 votes]: If $M$ is a hyperfinite type $I\!I\!I_1$ factor, then (at least conjecturally), its group of outer automorphisms is a $K(\mathbb Z,3)$.
This is based on the following three properties of that von Neumann algebra:
• The group of unitary central elements of $M$ is a circle, and thus a $K(\mathbb Z,1)$.
• The group of unitaries in $M$ is contractible.
• The automorphism group of $M$ is contractible (conjectural).
To see that $Out(M)\cong K(\mathbb Z,3)$, apply the long exact sequence of homotopy groups to the following two fiber sequences:
$$
U(Z(M)) \to U(M) \to Inn(M)
$$
$$
Inn(M) \to Aut(M) \to Out(M)
$$
As a consequence, we also get that $BOut(M)\cong K(\mathbb Z,4)$.
I recommend my talk "A K(ℤ,4) in nature"  (MSRI, April 2014), for an explanation of how to realize $Out(M)$ as the automorphism group of a naturally occurring mathemtical object.