## 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: 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.

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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 2010 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 2010 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 2010 at 16:29
Rourke and Sanderson's "Introduction to Piecewise-Linear Topology" is also out of print, and is near impossible to find. Milnor's "Whitehead Torsion" article is available on-line, is quite terse and contains much essential information. I imagine between a books like Milnor's h-cobordism notes, Kosinski's book and Milnor's Whitehead torsion notes you could put together a reasonable course on the s-cobordism theorem. – Ryan Budney Oct 27 2010 at 17:17
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 2010 at 18:10

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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