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?"
Best Answer
This is indeed a mystery. I presume the question refers to wedges of spheres of the same dimension, where there's a simple criterion (n-dimensional and (n-1)-connected, for some n). For wedges of spheres of different dimensions I don't know any such criterion. Even when it's known that a complex has the homotopy type of a wedge of n-spheres, it can be difficult to find cycles representing a basis for the homology. Proofs of (n-1)-connectedness are often by induction and hence not really very enlightening, in my experience with complexes arising in combinatorial low-dimensional topology. Ideally such a proof would proceed by showing that after deleting the interiors of some top-dimensional cells, the resulting subcomplex was contractible, hopefully by an explicit contraction. There's even one important case, Harvey's curve complex of a surface, where the complex has higher dimension than the wedge of spheres that it's homotopy equivalent to. It almost seems like a metatheorem in this area that any naturally-defined complex is either contractible or homotopy equivalent to a wedge of spheres. I can't think of any counterexamples, just off the top of my head. Perhaps in other areas the proofs are more enlightening.