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Background

The use of tools from algebraic topology to study simplicial complexes coming from point cloud data has been thoroughly discussed in the papers of Carlsson, Zomorodian, Ghrist, Edelsbrunner, Harer and more. Computing homology can be achieved in cubic time (in simplicies) and using persistent homology one can make an educated guess on which non-zero homology groups are noise and which represent n-dimensional holes in the point cloud.

What

In smooth theory one can use Morse theory to compute the homology of a closed manifold. One can also give a description of the cells in a CW complex which is homotopy equivalent to the manifold as well as computing the cohomology ring. I wonder what other "properties" of the manifold one can find using Morse theory.

Why

The idea is to use Forman's discrete Morse theory to calculate properties of simplicial complexes. Since it is computationally expensive to define a discrete Morse function - it has to be something else than homology.

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You might test for tightness of embeddings, but I guess you would most likely go the other way round, that is using (co-)homology to bound the number of critical points of suitable functions $M \to \mathbb R$. –  Alexander Thumm Mar 28 '11 at 21:40
    
A discrete Morse function gives you a homotopy equivalence to a CW complex with fewer cells, so it can contain much more information than just homology. –  Grumpy Parsnip Mar 29 '11 at 0:18

1 Answer 1

Morse theory is a very rich topic. Already, having the cohomology ring is a huge deal. That tells you a lot more about the manifold than just knowing the Betti numbers, for instance.

You can also use Morse theory to get at equivariant cohomology (in the case of a manifold with a group action) and apparently Steenrod squares (I went to a talk I did not fully understand in which the speaker claimed this).

There are some nice survey papers by Bott, notably Morse theory, old and new. There is another I am looking for. If I find it, I will update.

There is also a survey by Martin Guest arXiv:math/0104155v1.

Unfortunately, I can't speak to the computational aspects of any of these.

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