TITLE: Natural examples of finite dimensional spaces with interesting 2-type
QUESTION [5 upvotes]: Riemann surfaces provide interesting examples of 1-types - interesting as they have roles in diverse areas. However, apart from 2-dimensional lens spaces, I can't readily bring to mind natural examples of spaces with non-trivial first two homotopy groups (non-trivial firt $k$-invariant optional, I suppose). Given a crossed module one can get an interesting 2-type, but this is via geometric realisation, so hardly finite-dimensional and not smooth in the usual sense (maybe in some exotic notion of smoothness).
Do natural examples of spaces with interesting 2-types turn up anywhere?
I tried to find out if I could construct one in a naive way at this question, but it fell over.

REPLY [4 votes]: Cappell-Shaneson knots are a special class of knots in homotopy 4-spheres.  In a sense they were designed as an example of knots with a well-known but prescribed 2-type.  
The definition is like this.  Look at bundles over $S^1$ whose fiber is a once-punctured $(S^1)^3$.   The monodromy is an element of $A \in GL_3(\mathbb Z)$.  A non-trivial (but fun) exercise is to check this manifold $M = \left((S^1)^3\setminus\{*\}\right) \rtimes_A S^1$ is the complement of a smoothly embedded $S^2$ in a homotopy $4$-sphere if and only if $det(A) = \pm 1$.  
$\pi_2 M$ has a single generator as a module over $\pi_1 M$, but it's far from a free module over $\pi_1 M$.  Perhaps the best way to describe $\pi_2 M$ is that it's a Laurent polynomial ring $\mathbb Z[x^\pm,y^\pm,z^\pm]$ where the $x,y,z$ correspond to the generators of $\pi_1 ((S^1)^3 \setminus\{*\})$  i.e. this is $H_2$ of the universal cover, which has a natural identification with $(\mathbb R^3 \setminus \mathbb Z^3) \times \mathbb R$.  The action of $\pi_1 M$ is just the action on this covering space, so you get not just multiplication by units $x^ay^bz^c$ but also the automorphisms of $\mathbb Z^3$ (coming from $A \in GL_3(\mathbb Z))$ acting on the exponents of the polynomials. 
The homotopy 4-spheres that contain Cappell-Shaneson knots were once considered possible counter-examples to the smooth 4-dimensional Poincare conjecture.  Recent work of Akbulut and Gompf seem to have largely removed this possibility.