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](https://mathoverflow.net/questions/35847/interesting-representations-cohomology-of-surface-groups), but it fell over.



Natural examples of finite dimensional spaces with interesting 2-type
The 2-type of a 4-manifold is an extremely interesting invariant. In fact, work of Hambleton and Kreck shows that in many cases it determines the homotopy type (if one adds the intersection form as an obvious additional invariant). As a consequence, such 2-types have a very rich structure. 


It's quite tricky to figure out which 2-types appear in this way, just like it is tricky to figure out which groups arise as fundamental groups of 3-manifolds. 


This is a very interesting open problem, even if one puts aside the (difficult) 4-manifold questions and just works with 4-dimensional Poincare complexes.


