# Discrete Morse theory and existence of minimal complex

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.

• Have you read the Whitehead theorems on minimal CW-complexes? They're in many textbooks, for example, G.W. Whitehead's text, or section 4.C of Hatcher's notes. See the references in Hatcher's notes for more details. – Ryan Budney Oct 27 '10 at 14:20

• @JimConant That paper seems to have been written "too early". In particular, it contains the incorrect claim (Example 1.6) that the Whitehead group of $\mathbb{Z}\Pi$ for $\Pi$ a finite abelian group is trivial. The first counter-example is $\Pi = \mathbb{Z}/5$ whose whitehead group is $\mathbb{Z}/2$. – Vidit Nanda Sep 23 '14 at 3:58