- #1

lichen1983312

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I am self leaning some basic cohomology theory and I managed to go through from the definition to the universal coefficient theorem. But I don't think I get the main point of this theory, I like to ask this questions:

Is such an abstract theory practical?

I would say that homology is practical, because the chain groups could be built on the basis of maps from simplexes to the space, which is intuitive and easy to operate. Since the boundary operator also has clear geometric meaning, at least in theory one can just write down the chain group and follow a well defined procedure to compute the homology group.

For the cochain groups ##C_n^ * = Hom({C_n},G)## , the elements are just homomorphisms. This construction is too abstract that I cannot see what is this group look like and how is cohomology gorup ##\ker /im## computed. If one cannot easily do these things, why would we even need this theory ?

Is such an abstract theory practical?

I would say that homology is practical, because the chain groups could be built on the basis of maps from simplexes to the space, which is intuitive and easy to operate. Since the boundary operator also has clear geometric meaning, at least in theory one can just write down the chain group and follow a well defined procedure to compute the homology group.

For the cochain groups ##C_n^ * = Hom({C_n},G)## , the elements are just homomorphisms. This construction is too abstract that I cannot see what is this group look like and how is cohomology gorup ##\ker /im## computed. If one cannot easily do these things, why would we even need this theory ?

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