Why Is Cech Cohomology More Common Than Cech Homology?

[WWGD]

Hi All,

I am curious as to the reasons why one chooses covariant vs. contravariant theories; specifically, I see mention of DeRham Cohomology and Cech Homology, but I rarely see mention of the covariant counterparts DeRham and Cech homology theories.

I think one uses DeRham Cohomology , because it deals with differential n-forms, and n-forms pullback contravariantly, i.e., given a smooth map F: M-->N between manifolds, we get a pullback:

F* : N* -->M* , where N*, M* are the respective dual spaces of N, M. Something similar is the case for the double-, triple- , etc. duals, all of which pullback contravariantly.

Now, how to explain that Cech cohomology is more common than Cech homology? I guess this has to see with properties of sheafs. Now I know relatively little about sheaves. Is this the reason
for using cohomology? If not, what is the reason?

Thanks.

 

[Terandol]

I've never heard of de Rham homology but a version of Cech homology called strong homology which satisfies the Eilenberg-Steenrod axioms does exist (note that what is often simply called Cech homology does not satisfy these axioms) and is useful albeit usually in more exotic spaces (ie. things which aren't locally as nice as CW complexes.) Have you ever heard of shape theory? It is essentially an attempt to perform an analogue of algebraic topology on these kinds of spaces where the normal methods don't work well. A simple example is the Polish circle (put a copy of the topologists sine curve as a segment in a circle...see Exercise 7 of Section 1.3 in Hatcher for a picture) which has the same homotopy groups as a point in all dimensions but is not homotopy equivalent to a point. Strong homology is the homology theory defined in strong shape theory to deal with these types of spaces. This homology theory can be shown to agree with singular homology for nice enough spaces so it can be thought of as some sort of an extension of ordinary homology theory.

The reason that Cech homology isn't mentioned nearly as much is probably just that defining it in the obvious way, as the inverse limit of the simplicial homology of nerves of open covers, does not yield a proper homology theory in the Eilenberg-Steenrod sense. One way to think of the difference between Cech homology and Cech cohomology is that for homology you need to take an inverse limit and for cohomology you need to take a direct limit. Direct limit is an exact functor (at least in modules) so the exactness axiom for Cech cohomology will hold. On the other hand, inverse limits are only left exact but not right exact functors so taking inverse limits will not preserve exactness and the axiom fails. Strong homology fixes this by taking the homology of homotopy limits rather than the inverse limit of homology and turns out to preserve exactness (this is of course just a vague description of the idea. If you want the precise definition I would suggest the book 'Strong Shape and Homology' by Mardesic but it is a fair bit more involved than the usual constructions of Cech cohomology.)

So ultimately, I think it just boils down to convenience. It is much harder to construct a good Cech homology theory than it is to construct a good Cech cohomology theory (here I guess good just means it satisfies Eilenberg-Steenrod) so people tend to avoid it unless they are dealing with somewhat pathological spaces where ordinary Cech cohomology isn't very useful.

 

[WWGD]

Excellent , Terandol, very helpful , I will give it a read; I found an intro version in H&Y's Topology too.

 

[WWGD]

Sorry for my ignorant statement about "pulling back contravariantly"; pullbacks are _by definition_ contravariant, so saying something pulls back means it is contravariant ( as a functor) , and talking about pushforwards means the object is covariant as a functor.