# 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.

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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

This is not exactly what you asked, but it's certainly not the case that every CW complex has a discrete vector field where the Morse complex has trivial differential. In particular this would imply that chain complex is simple-homotopy equivalent to a chain complex with no differential. However, simple-homotopy equivalence is well-known not to generate homotopy equivalence. In particular, the Whitehead torsion is an obstruction which lives in the Whitehead group of the fundamental group. Marshall Cohen's book on Whitehead torsion is the canonical place to learn about this.

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Cohen's book is getting increasingly difficult to find. –  Ryan Budney Oct 27 '10 at 15:00
Yes, that's a shame. Do you know of a good reference for simple-homotopy theory that's easier to find? –  Jim Conant Oct 27 '10 at 16:29
One can find a pdf-file with Cohen's book on the web as well as Rourke and Sanderson's "Introduction to Piecewise-Linear Topology". –  Petya Oct 27 '10 at 18:10
@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 at 3:58
@ViditNanda: it was an early paper, but it is so well written that I still think it's worth reading. The incorrect claim you are talking about has a footnote explaining that the announced proof (by other authors) has serious problems. I am not aware of any mistakes within the paper itself. –  Jim Conant Sep 23 at 11:13

In this recent paper, Kozlov gives a sufficient condition for an acyclic matching on a CW complex to be a wedge of spheres (it is, unfortunately, not sufficient)

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One of your "sufficients" should be a different word? –  Jim Conant May 25 at 3:39
I think the second should be "necessary" instead, but cannot check as I don't have access at the moment. –  Camilo Sarmiento May 30 at 21:06