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:
- Simplical Homology
- Singular Homology
- 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?