Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
1  
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
add comment

2 Answers

up vote 3 down vote accepted

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.

share|improve this answer
2  
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
    
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 '10 at 17:17
1  
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
add comment

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)

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.