# homology groups, physical interpretation

One can roughly interpret Betti numbers as the number of n-dimensional holes of our space. Is there a reasonable interpretation for torsion coefficients? Moreover, for 'physical' applications, are they significant?

EDIT: let's use the more accurate vague phrase: "Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces," from wikipedia.

EDIT 2: In a quasi-hamiltonian formulation, the motion of (something) traces out various paths along a manifold which can be modeled as a superposition of a few dynamical systems (at least in this particular situation). We want to simplify the data that describes this motion while preserving some of the 'qualitative' features of the (discrete) topological space. I am certain that when processing the data, we will attempt to preserve the Bhetti numbers. Should I also attempt to preserve the torsion coefficients, if I am to study physical meanings?

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"Number of holes" is very vague. For example, do you feel like saying a Klein bottle has only one 1-dimensional hole, or two? $H_1 \simeq \mathbb Z \oplus \mathbb Z_2$. For your latter question, could you supply context for what you consider physically significant about Betti numbers? –  Ryan Budney Aug 19 '11 at 23:04
I'll edit the post. Let's replace number of holes by "number of unconnected n-dimensional surfaces." The context is roughly that I will be computing (approximate) homology groups for a manifold defined by a point cloud. In lower dimensions, I could visualize what the Betti numbers are measuring, but I'm not sure about the torsion coefficients. –  Tom S Aug 19 '11 at 23:19
Your intuitive notion of Betti number is technically wrong. For example, $\mathbb RP^2$ has an embedded circle which once you cut along it, you're left with a disc, yet $H_1 (\mathbb RP^2) \simeq \mathbb Z_2$. Why don't you consider this example and the Klein bottle example ? –  Ryan Budney Aug 19 '11 at 23:40
Also, it's not clear how any of this is "physical". Perhaps you're looking for an "intuitive" or "visual" understanding of these concepts. Usually when one says "physical" one means an interpretation in terms of some strict mechanical type model, and in this regard none of the above qualifies. –  Ryan Budney Aug 19 '11 at 23:45
What dynamical system are you studying, and what do you know about the Betti numbers -- relevant to the dynamics? Feel free to add it to your question. –  Ryan Budney Aug 20 '11 at 0:29