# Visualized definition of cohomology

I cannot imagine how cohomology is related to graph theory, actually I read solid definition from wiki, and to be honest, I cannot understand it. e.g I know what is homology (in simple term), group of functions such that I can continuously convert each of them to another one, but, is there similar visualization method for cohomology? (I'm not looking for exact definition, I want to imagine it, actually this is in graph theoretic concept). for more information see introduction of this paper. I want to understand it in this paper, how is useful? how to imagine it?

P.S1: my field is not related to group theory, and as in introduction author wrote, this paper doesn't need deep group theoretic definition! and I don't want to be deep in group theory. Just looking for simple way to understand them.

P.S2: I think I can imagine what is free group (which is in introduction of paper), at least by Calay graph seems to be easy to imagine it.

• You can 'think' of cohomology as 'dual of chains'. When you have linear transformations of vector spaces, you can consider its dual transformation (in linear algebra). So in homology theory you can do 'something similar'. Note I'm using quotes many times. It is just an idea. – Sigur Jan 22 '13 at 11:07
• This may have something to with my answer here: math.stackexchange.com/questions/225938/… – Aaron Mazel-Gee Jan 24 '13 at 7:10

## 1 Answer

I think it's fair to see to say that, in general, cohomology is not used in graph theory, and so your question does not have an answer.

Schrijver's paper is one of the few exceptions, and he states that he is just using the language. So I think you best advice is read some introductory texts to algebraic topology. I say "some" because the introductory texts tend to treat cohomology or covering spaces/free groups, but not both. Note that covering spaces do turn up regularly in graph theory, see e.g. voltage graphs.

• So you think is harder than express it in simple way? is there anything like homotopy for homology, to make it easy to understand cohomology? Actually I read a basic topology book (not entirely, but most parts), but I couldn't find cohomology on that book. – Saeed Jan 22 '13 at 13:26
• I think the best way to get a feel for cohomology is to look for a basis text that treats it. If you are comfortable with homology, then I think you will not find this difficult to learn. The essence of my response was that there is no "natural connection" between graph theory and cohomology. – Chris Godsil Jan 22 '13 at 13:51
• I see your point, I can't say I understand homology well, but is easy to understand homotopy so for surfaces is enough, but for higher dimensions need more intuition. but I cannot get cohomology even for 2 dimensional space, I cannot convert solid definition to something for imagine. – Saeed Jan 22 '13 at 14:12
• @Saeed: It's probably not a good idea to assume that you understand homology just because you understand homotopy. The two are related, but it's probably fair to say that homotopy is equally closely related to cohomology. – Noah Stein Jan 23 '13 at 14:20
• @Saeed: Why does the prefix "co-" make you think that the admittedly-vague picture you have of the relationship of homotopy to homology does not apply equally well to cohomology? Homology is much more closely related to cohomology than to homotopy. On an intuitive level, homology and cohomology carry the same kind of information and the definitions are very similar. The advantage of one over the other in a given application mostly amounts to certain algebraic or category-theoretic properties simplifying various arguments. – Noah Stein Jan 24 '13 at 19:43