Let me try and put the question in context. I am studying certain subsets of the tangent bundle of a sphere. I also have a regular CW complex which is a deformation retract of such a subset. Hence I have a description of the cells and the information that tells me which cell is in the boundary of which other cell. Fortunately this cell complex is a homotopy colimit of a diagram of spaces. As a result I can compute the cohomology groups (but not the product). 


All my examples concerning $S^1$ and $S^2$ show that these subsets have the homotopy type of wedge of copies of $S^1$ and $S^2$ respectively. Hence I am trying to prove that this is the case in all dimensions. In this process the only thing I was able to prove that there is a retraction from these subsets to the underlying sphere. 


So I would like to know about various methods to show that a space is a wedge of spheres. 


I understand that this question might sound vague and the information too little.



How to show that a space has the homotopy type of wedge of spheres ?
To follow up a bit on Mikael's answer, the notion of non-pure shellability is probably more relevant to your situation. Shellable simplicial complexes are wedges of spheres of equal dimension, but non-purity allows different dimensional spheres. You should look at papers by Michelle Wachs and Anders Bjorner if you're interested. However, this will require finding a simplicial decomposition of your space, which may be a challenge.


Added: Since this is now the accepted answer, I figure I should give the precise references. Both papers are on JSTOR (follow the links).


Björner, Anders; Wachs, Michelle L. 
Shellable nonpure complexes and posets. I.  

Trans. Amer. Math. Soc. 348 (1996), no. 4, 1299–1327.
<http://www.jstor.org/stable/i311403>


Björner, Anders; Wachs, Michelle L.
Shellable nonpure complexes and posets. II.
Trans. Amer. Math. Soc. 349 (1997), no. 10, 3945–3975. 
<http://www.jstor.org/stable/i311413>


