# Does every (co)homology functor (in particular, stable homotopy) factor through chain complexes?

Ordinary homology and cohomology factor through chain complexes via singular homology and cohomology. What about other (co)homology theories?

That is, for each spectrum $E$, do we have a lift in the following diagram?

$\begin{array}[ccc] & \mathsf{HoTop} & \overset{E}{\to} & \mathsf{GrAb} \\ & \underset{?}{\searrow} & \uparrow \\ & & \mathcal{D}(\mathsf{Ch}_{\mathbb{Z}}) \end{array}$

Where $\mathsf{HoTop}$ is the homotopy category, $E$ is $E$-homology or $E$-cohomology (in which case it's contravariant, of course), $\mathsf{GrAb}$ is graded abelian groups, $\mathcal{D}(\mathsf{Ch}_{\mathbb{Z}})$ is the derived category of chain complexes in abelian groups, the functor $\mathcal{D}(\mathsf{Ch}_{\mathbb{Z}}) \to \mathsf{GrAb}$ is homology, and the functor labeled "?" is the desired lifting.

It would just about suffice to do this in the universal example of stable homotopy ("just about" only because we have to do this for all spectra now), because we have a factorization:

$\begin{array}[ccc] & \mathsf{HoTop} & \overset{E}{\to} & \mathsf{GrAb} \\ \downarrow & \underset{\pi^S}{\nearrow} \\ \mathsf{HoSp} \end{array}$

where $\mathsf{HoSp}$ is the stable homotopy category, $\pi^S$ takes stable homotopy groups, and $\mathsf{HoTop} \to \mathsf{HoSp}$ is either $E\wedge(\Sigma^\infty-)$ (for homology) or $\operatorname{Fun}(\Sigma^\infty -,E)$ (for cohomology).

So in some sense I'm asking about the relationship between two sorts of graded-abelian-group-valued decategorification: taking homotopy groups of a spectrum, and taking homology groups of a chain complex.

• The map ${\cal D}(Ch_{\Bbb Z}) \to GrAb$ has a section, sending a graded abelian group to the chain complex with zero differential, and so in a very weak sense the answer to your question is "yes". However, this doesn't respect any of the ambient structure (such as the triangulated structure in the derived category). – Tyler Lawson Oct 2 '15 at 5:57
• The relationship between taking homotopy groups of a spectrum and taking homology groups of a chain complex is given by the stable Dold-Kan correspondence, which identifies chain complexes of abelian groups with $H \mathbb{Z}$-module spectra (in a way that sends homology to homotopy). – Qiaochu Yuan Oct 2 '15 at 6:35
• There is a long ago published paper to the effect that the only homology theories computable by chain complexes are products of ordinary homology theories, and there is a follow up paper (or two?) by a different author that gives a modified notion of chain complex for which there are more theories, but I can't remember authors or dates. For sure these papers exist though! – Peter May Oct 3 '15 at 20:29

No (I mean, not in a triangulated way), otherwise any generalized homology theory of a mod 2 Moore space would be 2-torsion, but this is not true for mod 2 stable homotopy groups (it's well known that you get a cyclic group of order 4). For positive results under extra hypotheses see:

Heller, A., 1966. Extraordinary Homology and Chain Complexes, in: Proceedings of the Conference on Categorical Algebra, La Jolla. pp. 355–365.

Neeman, A., 1992. Stable homotopy as a triangulated functor. Invent Math 109, 17–40. doi:10.1007/BF01232016

• I'd edit these links to give human-readable references, but I'm short on time. – David Roberts Oct 2 '15 at 6:39
• When Serre spoke in Harvard's Basic Notions seminar on sheaf theory (in the late '90s), this was essentially the example he gave to mollify Bott, who had stood up to declare the lecture over once derived categories were so much as mentioned. – Allen Knutson Oct 2 '15 at 11:52
• 1. Thanks, this is just what I was looking for! 2. The fact that my library has a print copy of the La Jolla proceedings but Springer still wants to charge me for an electronic copy gets me every time. 3. It seems strange to me that it's actually possible to do this 2-locally, given that it's certainly not something that's true for "formal" reasons. – Tim Campion Oct 2 '15 at 13:09

As written by Fernando, the answer is no if you suppose that you have exactness fo pairs at the chain level. In fact for a functor $$L_*:CW^{pairs}\rightarrow Ch(Ab)$$ if you have a short exact sequence of chain complexes $$0\rightarrow L_*(A,\emptyset)\rightarrow L_*(X,\emptyset)\rightarrow L_*(X,A)\rightarrow 0$$ and if the homology $$\mathcal{L_*}=H_*(L_*(-))$$ is a generalized homology theory then we have:

Theorem (Burdick, Conner, Floyd 1968).To each finite CW-pair (X,A) we have a natural isomorphism: $$\mathcal{L}_n(X,A)\cong \bigoplus_{p+q=n} H_p(X,A;\mathcal{L}_q(pt)).$$

Ref: "Chain Theories and their Derived Homologies", Proc . Amer. Math. Soc. l9 (l968) Proceedings of the AMS (1968)

• I'd edit the link to make this a link to the abstract, and make it human readable, but I'm short on time. – David Roberts Oct 2 '15 at 6:40
• @DavidC , thanks for writing this example! I knew the paper but I didn't remember the hypotheses and didn't find it in a couple of attempts. It's interesting to compare with the results of Heller and Neeman I link to. They relax the condition of strict short exact sequences of complexes to exact triangles in the derived category. In this way they get many more examples of factorizations, including stable homotopy groups after inverting 2, for which the Atiyah-Hirzebruch spectral sequences does not collapse. – Fernando Muro Oct 2 '15 at 11:39