TITLE: Homotopy type of non-Cohen-Macaulay complexes
QUESTION [9 upvotes]: Most interestingly defined (pure) simplicial complexes that occur in topological combinatorics are Cohen-Macualay. Some of these are even shellable (or have the homotopy type of a wedge of spheres; of same dimension). 
I would like to know examples of pure simplicial complexes (whose simplices are combinatorially defined) that are not shellable. 
Moreover, can anybody point out papers in which author(s) deal with simplcial complexes that do not have homotopy type of a wedge spheres?

REPLY [2 votes]: Here are some examples:
There are many papers that deal with topological properties of these complexes and they are rarely Cohen-Macaulay.
1) Triangulated manifolds are very interesting combinatorial objects, and usually, unless the manifold is a sphere or a ball they are not Cohen-Macaulay. Many aspects of the enumerative theory of face numbers of spheres have interesting generalization to manifolds. There is also rich examples of triangulations of specific manifolds with few vertices.
2) Within triangulations of manifolds there is a special role to $d/2+1$ neighborly manifolds when $d$ is even. Their existence for $d=2$ is the Heawood conjecture. For larger values of $d$ only a few examples are known like the Kuhnel 9-vertex triangulation of $CP^2$.
3) Triangulated pseudomanifolds (every co-dimension 1 face is included in two maximal faces + some connectivity assumption) can also be fascinating combinatorial objects. 
Here are three more interesting examples.
4) Triangulations of algebraic varieties (with singularity) are very interesting combinatorial objects.
5) Quotients of buildings.
6) High dimensional expanders.