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: 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.