A minimal complex is a CW complex whose only cells are the homology cells.

Is there some sort of criterion on CW complexes about existence of minimal complexes?

Actually I am working on a problem of understanding homotopy type of certain spaces (see: http://mathoverflow.net/questions/43711/how-to-show-that-a-space-has-the-homotopy-type-of-wedge-of-spheres)

My hope was to use discrete Morse theory (acyclic matching of face poset to be precise) and find the minimal complex. But then I don't know if the existence of the minimal complex is always guaranteed.