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Do different cohomology theories essentially just exist because there are distinguished homology theories associated with them? If yes, is it known if there is always a relation like the Poincaré duality? If no, is there such a thing as a universal cohomology, where you can map down to specific ones.

In the same line of thought, how many diffent homologies can be defined for a certain compex and how many complexes can be defined for a certain topology?

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Asking more focused questions is generally a good idea. An answer to what you wrote is essentially a survey on the subject... – Mariano Suárez-Alvarez Jan 6 '12 at 1:50
@Mariano Suárez-Alvarez♦: Okay, I edited away the extraordinary part. The first part might be answered by a yes/no, hopefully with some comments. – NikolajK Jan 6 '12 at 1:55
One thing I've heard in extremely anecdotal circumstances is that in some sense the number of extraordinary cohomology theories, seemingly like every other "extraordinary" piece of any classification theory, is related to ADE classification. I'd love to hear somebody clarify/expand on that. – NKS Jan 6 '12 at 2:54
If, by cohomology theories, you mean those which satisfy the Elienberg-Steenrod axioms, then there is only one set of cohomology groups up to isomorphism (for well behaved spaces). The many models (simplicial, CW, Morse, Čech, deRahm, sheaf) come from the several ways of describing spaces and the interesting objects defined on them. – Adam Jan 6 '12 at 2:56
up vote 5 down vote accepted

This is a really good question, and I think Akhil does a great job answering. I just wanted to add a bit.

So just like Akhil points out, cohomology theories are really spectra in disguise and as he states for every cohomology theory there is a homology theory. I am not aware of any of these homology theories that can be described as the homology of an associated complex other than ordinary homology. For example, I don't think anyone knows how to associate a complex to a space so that the homology of it is the K-theory of the space. So this is not how they come about, but the story is more interesting in fact.

They come from Geometry more or less, or at least that is where they started coming from. Things like (co)bordism and K-theory come from Geometry. Bordism is about manifolds that sit over the space you are interested in. K-theory is about vector bundles over your space. These are both obviously very geometric objects, the fact that homotopy theory can be used to study this is, to me, quite beautiful. These were the first two homology theories that people recognized after ordinary homology. They don't happen to satisfy one of the Eilendberg-Steenrod axioms, the one about the value on a point being concentrated in degree 0. These two theories started a lot of interesting things.

Other cohomology theories, as far as I know, are still lacking in geometric interpretations though. What I mean by that is we think of $\alpha \in K^0(X)$ as being an equivalence class of vector bundles over $X$ (when $X$ is nice enough). We can do something similar for $MG_*(X)$ but I am afraid I will get some dimensions off (here $G$ is the structure group of the stable normal bundle of the type of manifolds you are thinking about over $X$).

We can construct a lot of other cohomology theories though. They mostly come from looking at $MU$ which is universal in the sense you ask about, although not quite. $MU$ is universal among all complex oriented cohomology theories. A cohomology theory $E^*$ is complex oriented if it has a well behaved choice of Thom class $u(\xi) \in E^*T(\xi)$ for $\xi$ a complex vector bundle over $X$ ($T(\xi)$ is the Thom space of the complex vector bundle $\xi$). If we want to think about spectra, we would just say that there is a map of ring spectra $MU \to E$. I know some of this seems like jargon, and I would not worry about digesting it now, just know that such a thing does exist when you add different adjectives. However, I am not sure about the relation to Poincare duality, although I am sure it is relevant.

K-theory is something I would suggest you look at if you feel really comfortable with de Rham theory. I would recommend looking at May's Concise Course, he has some really well laid out stuff on K-theory and cobordism. It moves pretty quickly though.

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Turns out Akhil had no problem getting it all right in the following answer:… – Sean Tilson Jan 6 '12 at 5:26
Thank you for the kind words. I just want to point out (for the benefit of the OP) that there is a somewhat uninteresting variant of ordinary cohomology where instead of taking $\mathrm{Hom}_\bullet(C_\bullet(X), G)$ for $G$ an abelian group, $G$ is replaced by another complex; this is represented by the generalized Eilenberg-MacLane spectrum (which can be constructed by inverting Dold-Kan on $G$, if it is nonnegatively graded at least). – Akhil Mathew Jan 6 '12 at 16:28

I think the more fundamental notion, from the point of view of homotopy theory, is that of a spectrum. I will think of a spectrum naively, as a sequence of pointed spaces $X_n$ together with homotopy equivalences $X_n \simeq \Omega X_{n+1}$. In this case we can construct a cohomology theory on (nice) pointed spaces $A$ given by $A \mapsto [A, X_n]$ (in degree $n$), and a homology theory via $\varinjlim \pi_n(A \wedge X_{m+n})$. Any homology or cohomology theory comes from a spectrum, by the Brown representability theorem, but the spectrum is generally not unique (at least in the case of homology), because of the existence of so-called "phantom maps." However, you can do homotopy theory with spectra, e.g., make them into a nice and category with a fair bit of structure such as homotopy colimits and exact triangles.

There are forms of Poincaré duality that apply to generalized cohomology theories arising from ring spectra, under appropriate hypotheses: one needs the condition of orientability of the manifold (with respect to the ring spectrum). In this case, the statement of duality is quite similar to the usual one; see Adams's blue book, part 3 for a description.

(Incidentally, de Rham cohomology is not a "cohomology theory" in the usual sense of the word because it is usually only defined on smooth manifolds; insofar as it is, it is only ordinary cohomology with coefficients in $\mathbb{R}$.)

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This is a bit off-topic, but if I want to understand more about these stuff (motivation for spectrum, Poincare duality etc), where should I look? – Soarer Jan 6 '12 at 3:08
@Mariano: You are right; I have fixed the notation. – Akhil Mathew Jan 6 '12 at 3:09
@Soarer: I'm not really the right person to ask for this, but Adams's blue book is the canonical resource. I have also found Switzer's book on algebraic topology and Kochman's book on bordism to be quite helpful. I am told that Ravenel's green book is the book to study when you really want to go deep into the computations, but I haven't read much of it. – Akhil Mathew Jan 6 '12 at 3:11
Switzer is pretty amazing (if a bit dry... :) ) – Mariano Suárez-Alvarez Jan 6 '12 at 3:12
I'm guessing that the blue book is Stable homotopy theory? Thanks anyway! – Soarer Jan 6 '12 at 3:15

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