Decoupling the algebra from the topology in cellular homology

Question

The story of singular homology is pretty straightforward. One starts by constructing the singular chain complex functor $S : \mathbf{Top} \to \mathbf{Cha}$ (category of chain complexes with chain maps). Taking homology is then a purely algebraic operation happening in $\mathbf{Cha}$. The topological content of the theory is encoded in the properties of $S$. Here are a few questions about this view:

• Does S commute with homotopy colimits?
• Can the Eilenberg–Steenrod axioms be translated to conditions on the functor $S$?

The situation with cellular homology seems more convoluted. Consider the subcategory $\mathbf{CW}$ of filtered pointed CW complexes with filtration preserving maps. It can be realized as the category whose objects are acsending sequences of skeletons ($X^{-1} \to X^0\to X^1 \to \dots$) where $X^{-1}$ is a one point corresponding to the basepoint of the complex and whose arrows are collections of maps for every nonzero skeleton ($f_n:X^n \to Y^{n}$) s.t. the relevant diagram commutes.

So far all the constructions for cellular homology I've seen rely on ad hoc constructions via diagram chasing at the level of homology groups. It gets especially crude when the time comes to define a suitable boundary operator $\partial :H_{n+1}(X^{n+1},X^n) \to H_n(X^n,X^{n-1})$ . At which point there's an unavoidable digression into (not obviously canonical) ad hoc diagram chasing, "exact sequence of triples" and some more algebraic mess.

I do get the gist of it all. The fact that the characteristic maps give you maps between wedges of spheres at every level of the filtration and that these map induce maps on their top homology which is why the degree comes up. But i'm trying to get a simple formal picture of what's happening and it gets pretty discouraging.

Is there a construction analogous to the above construction for the singular case which uses a minimal amount of homological algebra and defines a functor $W: \mathbf{CW} \to \mathbf{Cha}$ which takes $CW$ complexes to chain complexes who can then further be manipulated algebraically to get the homologically equivalent cellular chain complexes

$$\dots \to H_n(X_n,X_{n-1}) \to H_{n-1}(X_{n-1},X_{n-2}) \to \dots $$

And that once wev'e passed to $\mathbf{Cha}$ this "manipulation" envolves only homological algebra?

Or maybe this is the wrong question entirely and something more subtle is happening beneath the surface? Some intermediate step which prevents a "decoupled" description?

Do I have to understand general spectral sequences of filtered spaces to really get the overall non ad-hoc approach to this?

Answer 1

Q1. Morally yes. You should think of singular chains as describing, loosely, the "free chain complex" on a space. In invariant language there is an $\infty$-category of spaces $\text{Space}$ and an $\infty$-category $\text{Ch}(\mathbb{Z})$ presented by chain complexes of abelian groups (one name for this is the $\infty$-category of "$H\mathbb{Z}$-module spectra," but you don't need to know this). There is also a forgetful functor

$$\text{Ch}(\mathbb{Z}) \to \text{Space}$$

and an invariant version of singular chains gives its left adjoint (in the $\infty$-categorical sense). Any left adjoint preserves homotopy colimits.

Q2. The Eilenberg-Steenrod axioms for singular homology can be reformulated as saying that singular chains, as an $\infty$-functor $\text{Space} \to \text{Ch}(\mathbb{Z})$, is determined by the fact that it preserves homotopy colimits and takes value $\mathbb{Z}$ on the one-point space. This is describing a universal property of $\text{Space}$ as an $\infty$-category: it's the free homotopy cocomplete $\infty$-category on a point, in the same way that $\text{Set}$ is the free cocomplete category on a point.

The Eilenberg-Steenrod axioms for extraordinary homology theories work the same way, but the target $\infty$-category has to be modified to be spectra.

Q3. The invariant content of CW complexes is that they describe spaces by building them up using iterated homotopy cofibers. When you pass this fact through singular chains, you get that you can describe singular chains on a CW complex by taking iterated mapping cones. When you do this you should get more or less the cellular complex, although I haven't checked the details.

Answer 2

the answers to your first questions is no - the singular complex of any kind of homotopy colimit involves singular simplices that do not respect any of the structure of your favorit homotopy colimit construction. So the resulting complex will be much larger than the homotopy colimit of the singular chain complexes.

Regarding the second question, I would see this differently. The singular simplicial complex $S(X)$ has a geometric realisation $|S(X)|$. This is a CW complex (sorry about that) that comes with a natural map $p\colon |S(X)|\to X$. By a theorem of Milnor, this map is a weak equivalence (it induces isos on all homotopy groups $\pi_k$). If you are willing to add an "invariance under weak equivalences" axiom to the Eilenberg-Steenrod axioms, than this tells you that $p_*\colon H_*(|S(X)|)\to H_*(X)$ is an isomorphism (similarly for cohomology). Now, the (co-) homology of CW complexes is entirely determined by the classical Eilenberg-Steenrod axioms, if you add an "additivity axiom" which says that the (co-) homology of a wedge of spheres is the direct sum (product) of their individual (co-) homologies. If you apply the construction of cellular (co-) homology to $S(X)$, you get the singular (co-) chain complex and arrive at the well-known formulas for simplicial singular (co-) homology.

To summarize: If you assume that your favourite (co-) homology theory satisfies the classical Eilenberg-Steenrod axioms, and additivity, and invariance under weak equivalences, then this (co-) homology is naturally isomorphic to singular (co-) homology.

If you don't like the text book approach to cellular (co-) homology, then you should at least try to understand that the construction gives you a natural isomorphism between "your" (co-) homology of a CW complex, and the cellular one. In particular, all the diagram chasing, ad hoc though it may seem, is completely natural in the technical sense.

If you want to learn spectral sequences, you can try to compute other, generalised cohomology theories (like $K$-theory) for CW complexes. However, there are some technical problems with spectral sequences that prevent you from getting a complete description. Because of the dimension axiom, the spectral sequence computations become much easier for ordinary (co-) homology, and they give you a complete answer. If you go through them, you arrive exactly at the diagrams that you know and love (but you will know why they look the way they do).

Answer 3

This is some sort of answer to your question: Do I have to understand general spectral sequences of filtered spaces to really get the overall non ad-hoc approach to this?

A general context for this area is given in the book titled Nonabelian algebraic topology:filtered spaces, crossed complexes, cubical homotopy groupoids (EMS Tracts vol 15, 2011) (pdf available). Spectral sequences do not occur in the book. However (strict) colimit arguments are used extensively, made possible by a Higher Homotopy Seifert-van Kampen Theorem (HHSvKT) for a homotopically defined functor $\Pi: FTop \to Crs$ from the category of filtered spaces to that of "crossed complexes", a kind of generalisation of chain complexes of abelian groups. This idea goes back to Blakers (1948) and J.H.C. Whitehead (1949), "Combinatorial Homotopy II". An immediate Corollary of the HHSvKT is that for a CW-filtration, $X_*$, $\Pi X_*$ is free on the characteristic maps of the cells. Another Corollary is a version of the Relative Hurewicz Theorem, describing $\pi_n(X \cup CA, x), n \geqslant 2$, when $(X,A,x)$ is $(n-1)$-connected, without using singular homology. There are also nonabelian calculations of second relative homotopy groups.

However the proofs use extensively cubical higher homotopy groupoids of filtered spaces; these are strict higher groupoids, and cubical methods are essential for the proofs, since they allow for "algebraic inverses to subdivision".

There are interesting relations between crossed complexes and chain complexes with a group(oid) of operators, which enable relations with cellular chain complexes of universal covers. The book gives some relation to history and gives an account of intuitions; there is more background in presentations on my preprint page.

A key part of this work is not so new, and dates back to two papers by Philip Higgins and me in JPAA, 1981.