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What does a dimension of a homology group tell us? In particular, suppose we form an arbitrary simplicial complex $S(G)$ from a simple graph $G$. Then we compute the homology groups of $S(G)$ and note the ones which are nonzero. What useful information can we glean by computing the dimension of the nonzero homology groups?

Edit. $S(G)$ is usually taken to be the coloring complex.

NebulousReveal
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    See the Wikipedia article on Betti numbers. Could you please clarify what you mean by "suppose we form an arbitrary simplicial complex from a simple graph" - well if the simplicial complex is arbitrary we will get an arbitrary result, won't we? – t.b. May 27 '11 at 17:33
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    Is your question about homology in general or about the homology of the coloring complex in particular? If the former, perhaps the discussion at MO might be useful: http://mathoverflow.net/questions/640/what-is-cohomology-and-how-does-a-beginner-gain-intuition-about-it – Qiaochu Yuan May 27 '11 at 18:39
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    Homology is an incredibly rich and vast field, and homology groups can tell you all sorts of things about different spaces. Without more detail about what you're looking for, we can't really give you more information. Since you seem to have a background in combinatorics, I'd say this question is a bit like asking, "Suppose I form a vertex set and edges from a topological space. What useful information can we glean from the resulting graph?" – MartianInvader May 27 '11 at 21:00
  • Why not try to answer the question yourself? If you want to start scratching the surface, compute the Euler characteristic of $S(G)$. This is readily done and expressible as a certain number that you can extract directly from the graph $G$. Is that number familiar to you? – Ryan Budney May 27 '11 at 21:54
  • I see it is the alternating sum of the Betti Numbers. – NebulousReveal May 27 '11 at 22:07
  • You might want to look at this question: http://math.stackexchange.com/questions/40149/soft-question-intuition-of-the-meaning-of-homology-groups/ – Josh May 28 '11 at 01:43
  • @PEV: that's only one of many equivalent definitions of the Euler characteristic. There's one that relates it directly to cell counts in your simplicial complex. If you go on to apply the definition of your simplicial complex you get a formula for your Euler characteristic directly in terms of the properties of $G$. – Ryan Budney May 28 '11 at 03:32

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