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I have decided to begin studying co/homology and I'm trying to work out the best approach to doing this. As I understand the situation, any system that satisfies the Eilenberg-Steenrod axioms qualifies as a Homology Theory. Specific examples of homology theory include:

  1. Simplical Homology
  2. Singular Homology
  3. Cubical Homology

This raises my first question: Which homology theory is best to start out with? Cubical homology seems nice and concrete and its easy to use it to calculate things. For this reason, it seems to me that this would be a good homology theory to learn for pedagogical reasons. Is this understanding accurate? Or, would it be better to simultaneously study, say, the three listed above? While cubical homology seems the easiest to learn, I'm not sure about its long-term value and whether I would be better off going the simplical/singular route. Finally, of the three homology theories above, are they equally "strong"? Are there things one can prove in the context of one but not in the other?

The other question I have regards how to approach cohomology. Since its effectively dual to homology it seems like it might be a good exercise to learn it simultaneously and treat stating/proving theorems in cohomology as exercises to reinforce homology theory. So, would it be better to learn homology completely and then go through cohomology or to learn the two theories simultaneously?

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Regarding cubical homology, see this MO thread. For my part, I don't really run into cubical homology very much, but I'm not a topologist. It seems that if you learn one theory then the others are be pretty easy to absorb. – Dylan Moreland May 11 '12 at 2:15
@DylanMoreland Very relevant thread; thanks for pointing it out – ItsNotObvious May 11 '12 at 2:18
There is no real choice between those three options... I would say that your question only shows you do not know a lot about algebraic topology :) Just pick a good textbook and read it through—all you ask here will answer itself as soon as you learn what it is you are asking! A request for recomendations of what is a good textbook might be an immensely more useful thing for you to ask if you really want to learn algebraic topology. – Mariano Suárez-Alvarez May 11 '12 at 2:22
(the difference between simplicial, singular, cubical, cellular, and many other variants is just technical—making distinction between them at a point where you are just starting is really not useful) – Mariano Suárez-Alvarez May 11 '12 at 2:24
To me, the difficulty in learning this material is to avoid getting bogged down in the technical aspects and actually learn some geometry. But someone will disagree :) – Dylan Moreland May 11 '12 at 2:37
up vote 6 down vote accepted

What about the de Rham approach to cohomology? The nice thing about this is that the complex of differential forms is very concrete and familiar if you've been exposed to at least multivariable calculus. You also avoid any difficulties that arise with torsion.

Another benefit is that the pairing of cohomology with homology is also already familiar: it is just integration.

A good source for this is Bott and Tu's ``Differential forms in algebraic topology."

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+1 I give anyone a point for recommending Bott and Tu. It really is one of the great classic textbooks of mathematics. : ) – Mathemagician1234 May 11 '12 at 3:58
Thanks for your constructive suggestion – ItsNotObvious May 13 '12 at 3:07

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