15
$\begingroup$

I am trying to understand whether there is a sense in which cohomology always relates to topology or whether this is the case only in particular examples. According to the Wikipedia page, a cochain complex is defined as: 

“… a sequence of abelian groups or modules ..., $A_0$$A_1$$A_2$$A_3$$A_4$, ... connected by homomorphisms $d^n$ : $A^n$ → $A^{n+1}$ satisfying $d^{n+1} \cdot d^n$ = 0. “

Based on this one defines the nth cohomology group $H^n$ as the group

$H^n := ker \, d^n / \, im \, d^{n-1}$ .

As a physicist, I am most familiar with examples of cochains and cohomology appearing in the context of topology. For instance, de Rham cohomology in which $d$ is identified with the exterior derivative and $A^n$ with the space of differential forms of degree $n$ on a smooth manifold $M$. In this case cohomology groups capture topological invariants of the manifold $M$.  

Although the general definition above makes no explicit reference to a topological space, I often see the claim that cohomology is a branch of topology. In fact, from the mathoverflow page cohomology is defined as:

"A branch of algebraic topology concerning the study of cocycles and coboundaries."

Thus, my question is the following:

Q: Is there a sense in which, given a general cochain complex, its cohomology captures topological invariants of some underlying topological space?

Q': If so, what defines the underlying topological space? 

Q'': If not, is there still a sense in which cohomology groups are some sort of “invariants?”

$\endgroup$
2
  • 17
    $\begingroup$ The homology groups of any chain complex $C_*$ are isomorphic to the simplicial homotopy groups of the simplicial abelian group $A_\bullet$ associated to $C_*$ by the Dold-Kan Theorem. These are in turn isomorphic to the homotopy groups of a topological space, namely the geometric realisation $|A_\bullet|$. $\endgroup$ Jun 9 at 9:48
  • 7
    $\begingroup$ @NeilStrickland But only for positively graded chain complexes, if everything is unbounded the connection with topology is more complicated (through spectra I guess). $\endgroup$ Jun 9 at 10:03
23
$\begingroup$

While it is always possible to introduce topology, it is not always the obvious or most useful thing to do. So at least in this sense, there are notions of cohomology that do not immediately connect to topology.

As an example group (co)homology, Lie algebra (co)homology, Hochschild (co)homology, ... all appear in algebraic contexts without necessarily having topological interpretations all too closely connected to them. There are of course topological interpretations, e.g. group cohomology is the topology of the classifying space of the group, but that's not necessarily the most useful way to view them. More useful often is the connection to representation theory (says the guy with a degree in representation theory) which comes from the view point that they're all derived functors on some module category.

EDIT: Because negatively-graded complexes were mentioned in the comments, let me just point you towards Tate cohomology which is a generalization of group cohomology to negative degrees.

$\endgroup$
9
  • 1
    $\begingroup$ Just "cohomology" in general? No. That encompasses so much that there is little to say without heavy use of category theory I think. Everything I know or have read about cohomology in general fundamentally requires category theory. $\endgroup$ Jun 9 at 14:07
  • 9
    $\begingroup$ I would go even further and say that talking about cohomology in depth requires category theory, period. That is what category theory was invented for. The moment when you look at more than one cohomology theory, you will ask yourself "why do 1 and 2 look similar?" and "how can I use that similarity in order to save on proving everything yet again the third time?". Both questions are answered by category theory. $\endgroup$ Jun 9 at 14:08
  • 1
    $\begingroup$ OK, thank you! By the way, I came across the nice book by Weibel "An Introduction to Homological Algebra" in case this is useful to others. $\endgroup$
    – Nuno
    Jun 9 at 14:38
  • 3
    $\begingroup$ I do like Weibel's book. @HollisWilliams, I think Weibel put an errata page on his web site. Do you mean that there are "many more" beyond that? $\endgroup$ Jun 9 at 20:17
  • 2
    $\begingroup$ Let me add Dolbeault cohomology of complex manifolds to the list. It contains subtle information about the complex structure. Of course, it can be turned into the cohomology of a certain sheaf on a topological space, but then the extra structure is hidden in that sheaf. $\endgroup$ Jun 10 at 10:15

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.