I got into a discussion today about how, just as with all the other ways of computing singular homology, there should be an internal sort of integration pairing for Cech cohomology with "Cech homology", whatever it is. My sense is that probably a Cech $n$-chain should be a formal sum of $(n+1)$-fold intersections, or maybe more precisely a formal sum of sections thereover. The boundary map should be determined by $\partial U_{i_0, \ldots, i_n} = \sum_\alpha (-1)^\alpha U_{i_0,\ldots,\hat{i_\alpha},\ldots,i_n}$, except for the serious problem that there's no way to extend sections. So perhaps instead my $n$-chains really should just be formal sums of intersections (with $\mathbb{Z}$-coefficients)? The abstract-ish reason for this is that abelian groups are just $\mathbb{Z}$-modules, and perhaps this is the same story as the pairing for nonorientable manifolds, where you twist one or the other (but not both) of homology and cohomology by the orientation sheaf. But there's an asymmetry here, because the definition of a sheaf makes it so that I have no choice but to twist my cohomology instead of my homology, and that feels wrong to me.
So, what's the right story? And assuming I've more or less got the right definition, how should I relate this back to singular homology?
Best Answer
There is a notion of Cech homology defined as Jim Conant says, but it is deficient as it does not satisfy many of the properties that one would like with a homology theory. If your spaces are at all general (i.e. more general than polyhedra, simplicial complexes etc) then Cech homology does not have long exact sequences in those situations that one would expect them. It also does not have really nice pairings with Cech cohomology. What goes wrong is that in taking the inverse limit of the homology of nerves of open covers one destroys exactness. (There are well known examples of very simple inverse sequences of exact sequences of abelian groups which when you take their limit end up being clearly not exact. The usual one relates to solenoid spaces.)
If you are just looking at manifolds or similar spaces, then you don't need to be too concerned about these phenomena, BUT if you are working with more general spaces, or with flows, dynamical systems etc. or are working on areas which interrelate with operator algebras, $C^*$-algebras, etc. then there are points to note.
There is a homology theory (Steenrod-Sitnikov homology or Strong Homology) which repairs the deficiencies of the Cech version. The idea can be summed up as saying first take the chains on the nerves of covers then form the homotopy limit of the result, finally take homology, so you replace `$lim H_n$', by $H_n holim$. This has good properties and does relate nicely to the $C^*$-algebra context, e.g. in the well known work of Brown Douglas and Fillmore from the 1970s. (References for Steenrod homology include the lecture notes of Edwards and Hastings, SLN 542, or Sibe Mardesic's book on Strong Homology.)