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Work related I have to deal with cohomology theory fairly soon. Unfortunately, I never had any classes on this, so I'd like to study it on my own. Before I dive into a book or two, I'd like to make sure that I have all required previous knowledge to actually understand it. It would be great if someone could give a list of topics one should chronologically cover in order to be prepared to attack cohomology. (In order to make sure that no topic is omitted, imagine this question is asked by a high-school student, who never had any advanced math.) Thanks for any suggestion!

Kagaratsch
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  • What type of math are you interested in? – Matthew Levy Jan 24 '15 at 20:07
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    From what I know so far, the cohomology needed will have to be applied in the context of geometry of vector spaces, symmetries, groups. I am not sure if this answers your question... – Kagaratsch Jan 24 '15 at 20:10
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    This highly depends on what kind of work you need to do and what you mean by "actually understand". If you only want it for the purposes of advanced calculus (de Rham cohomology), there are some reasonably self-contained books. Otherwise, a solid background in abstract algebra and topology will likely be required. Do you know what an exact sequence is? – snar Jan 24 '15 at 20:10
  • @Snarski: Oh, ok! Since you mentioned self contained books on de Rham cohomology, could you name an example, so I can take a look? I will make sure to look up abstract algebra and topology. – Kagaratsch Jan 24 '15 at 20:14
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    You can consider "Differential Forms with applications to the physical sciences" by Harley Flanders, Ch. 5. John M. Lee's "Smooth Manifolds" also has a more algebraic treatment of cohomology. One can find thorough information about abstract algebra in Dummit & Foote's "Abstract algebra"; cohomology is treated towards the end. Finally, I mentioned algebraic topology as homology takes on a very intuitive character there; Hatcher's freely available book can provide some guidance: http://www.math.cornell.edu/~hatcher/AT/AT.pdf – snar Jan 24 '15 at 20:53
  • Out of curiosity, what do you need cohomology for at work? – littleO Jun 26 '16 at 23:21
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    Madsen and Tornehave: From Calculus to Cohomology: https://www.amazon.com/Calculus-Cohomology-Rham-Characteristic-Classes/dp/0521589568 (specifically De Rham Cohomology) completely self-contained – Chill2Macht Jun 26 '16 at 23:26
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    You really must narrow the scope if you want a good answer, I think. – Hoot Jun 26 '16 at 23:35
  • You need a passion for holes. – Jacob Wakem Jul 03 '16 at 03:07

1 Answers1

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A. De Rham cohomology.

i) some general topology (basic notions: what topological spaces are, compactness, connectedness) ii) some smooth manifold theory (basic notions: what manifolds are, tangent spaces) iii) some linear algebra (basic notions: vector spaces, exact sequences, quotient spaces)

B. Sheaf cohomology.

i) some sheaf theory (basic notions: what they are, left exactness of the global sections functor) ii) some homological algebra (derived functors)

C. Singular cohomology i) some general topology (basic notions: what topological spaces are, compactness, connectedness) ii) some linear algebra (basic notions: vector spaces, exact sequences, quotient spaces)

These prerequisites are minimal in the sense that they allow you to understand the definitions, but you will of course need more to understand interesting/advanced results.

syzygy
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