# What's a cohomology that's not defined from a cochain complex?

According to Wikipedia:

In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex.

(emphasis mine)

When I read "often defined", I assume that to mean "not always". I've seen that Čech cohomology is defined from a cochain complex. de Rham cohomology the same. I checked a few more and all defined from a cochain complex. So what's an example of a cohomology that is NOT defined from a cochain complex?

• This is a good question. I don't know enough about the general theory to know of an example. But I actually think the quoted part has worse issues, namely that it claims that cohomology is something associated to a topological space, which is a way too narrow (or just really old) point of view. Jul 25 '16 at 8:07
• I can imagine that there are cases when a cohomology is defined in other terms, but one finds a cochain complex that may also be used to calculate it in order to justify calling it "cohomology". I can't think of any at the top of my head, though. Jul 25 '16 at 8:28
• You can always take as a cochain complex the complex having the corresponding cohomology groups as modules and $0$ as a differential. Jul 25 '16 at 9:01
• There is a relation between (generalized) cohomology theories and spectra. If you have a cohomology theory $h_n$ satisfying Eilenberg-Steenrod axioms, you get a spectra, that is a seq. of spaces $A_n$, with some relations. Then the homology groups $h_n$ of a space $X$ can be identified with the homotopy classes of maps into the the spectra $A_n$. I.e. you have an natural isomorphism $h_n (X) \cong [X , A_n]$. Jul 25 '16 at 11:21
• – paf
Jul 27 '16 at 13:13

Wikipedia is talking about generalized cohomology theories. They satisfy a bunch of axioms formally similar to what e.g. singular cohomology satisfies. But the so-called extraordinary cohomology theories (K-theory, cohomotopy, cobordism and so on) do not come from chain complexes. Indeed, by a theorem of Burdick--Conner--Floyd, in some sense if they did "come from chain complexes", then they would be given by a product of ordinary cohomology theories.

R. O. Burdick, P. E. Conner, and E. E. Floyd. “Chain theories and their derived homology”. In: Proc. Amer. Math. Soc. 19 (1968), pp. 1115–1118. issn: 0002-9939.

Roughly speaking, the theorem is that if you have a generalized cohomology theory $h$ given by the cohomology of a (functorial on pairs) cochain complex, and such that the long exact sequence comes from the long exact sequence of that cochain complex, then $$h^n(X,A) = \sum_{i+j=n} H^i(X,A; h^j(\text{point}))$$ is really given by singular cohomology with various coefficients.

But the emphasis here is on "cohomology theory". In a different setting (homological algebra), "cohomology" always means "cohomology of a cochain complex". Honestly, the Wikipedia article should probably be renamed to something like "Cohomology theories" or "Cohomology (topology)" because that's what it's really about.

There are various generalized cohomology theories which are typically not described in terms of cochain complexes, perhaps the simplest example being topological K-theory. There's a precise technical sense in which topological K-theory cannot be "described in terms of cochain complexes," but it's not easy to spell out.

• They are in fact never described by cochain complexes unless they're trivial (in a meaningful way), see my answer. Jul 27 '16 at 8:29
• @Najib: sorry, to be clear, "typically" modifies "described," not "various generalized cohomology theories." I just mean to point out that a priori, just because K-theory is not defined in terms of a cochain complex doesn't imply that it can't be. Jul 27 '16 at 8:34
• Hm, I guess I see your point: you can just take e.g. the cochain complex $K^i(X)$ with trivial differential? (Which of course cannot play nice with the l.e.s. due to the theorem I cite) Jul 27 '16 at 8:36

Our topology professor, Prof. Michael Weiss, defined in his lectures a homology theory for topological spaces (which I think he mentioned to be equivalent to Čech cohomology) by the means of something which he calls mapping cycles. These are elements of the sheafification of the free abelian group of continuous maps between any two topological spaces. Homology and cohomology are then defined as the groups of mapping cycles from and to spheres respectively, modulo homotopy and “constant” mapping cycles. There are no chain complexes involved.

Accidentally, there’s also a question here on math stackexchange which gives a short overview of the construction, namely here. The questioner also refers to the same lectures I refer to.

The lecture notes (in English) of his topology lectures in 2013/2014 and 2014/2015 at the University of Münster found on his web page give this definition of homology in chapter 5, section 3. Mapping cycles are introduced in chapter 4. Weiss calls it homology without simplices in contrast to the tradition of defining homology via simplices (and chain complexes) since Poincaré.

To my knowledge, the construction is due to Weiss himself.