TITLE: Why do wedges of spheres often appear in combinatorics?
QUESTION [43 upvotes]: Robin Forman writes in "A User's Guide to Discrete Morse Theory":

The reader should not get the
  impression that the homotopy type of a
  CW complex is determined by the number
  of cells of each dimension. This is
  true only for very few spaces (and the
  reader might enjoy coming up with some
  other examples). The fact that wedges
  of spheres can, in fact, be identified
  by this numerical data partly explains
  why the main theorem of many papers in
  combinatorial topology is that a
  certain simplicial complex is homotopy
  equivalent to a wedge of spheres.
  Namely such complexes are the easiest
  to recognize. However, that does not
  explain why so many simplicial
  complexes that arise in combinatorics
  are homotopy equivalent to a wedge of
  spheres. I have often wondered if
  perhaps there is some deeper
  explanation for this.

The question is: "Why so many simplicial complexes that arise in combinatorics are homotopy equivalent to a wedge of spheres?"

REPLY [15 votes]: Various simplicial complexes arising in combinatorics have the Cohen Macaulay property. Such complexes are always (homologically) a wedge of spheres of the same dimension (the dimension of the complex) and also locally have this property. Shellability is an important combinatorial property that again implies the complex and all links to be wedge of spheres. There are interesting extensions of these concepts to the case of spheres of different dimension. Shifted complexes are simplicial complexes whose vertices have a total order and with the property that if S is a face and R is a face of the same dimension with smaller vertices (namely there is a bijection a:R->S so that v<=a(v) for every v) then R is a face. Shifted complexes are always wedge of spheres (of different dimensions) and there are interesting operations which associate to arbitrary simplicial complexes, such shifted complexes. 
There are also quite a few interesting classes of simplicial complexes arising in combinatorics which are far from being wege of spheres. Take for example the chessbod complex whose faces are the locations on n nonattacking rooks in an n by n+1 chessboard board. for n=2 it is a hexagon for n=3 a torus for n=4 a certain pseudomanifold all links of vertices are tori, etc.