Singular analog of cellular homology

Question

Simplicial homology require that we cover the space X with a simplicial complex. But singular homology relaxes the requirement of such a discretization by considering all possible simplices in X. Although the later is not computationally favorable, it is helpful in proving many things.

Cellular homology, like simplicial homology, requires that we set up a CW structure on a space. Is there a homology theory analogous to singular homology that is made up of "singular cells" rather than singular simplices?

Answer 1

Just thinking on my feet: From what I can tell, the trouble with setting up a "singular cell" version of homology is that simplices, whether part of a simplicial complex or on their own, have boundaries that are themselves simplices. So the boundary of a singular simplex is a singular complex simply by restricting. On the other hand, the boundary formula for a cell in a cell complex is much more complicated, and taking the boundary of a (mapping of an) abstract cell doesn't continue to carry the same sort of information that the boundary of a (singular) simplex does. In other words, the same thing that lets us glue together CW complexes in ways that are more complex than what we do for simplicial complexes is a hindrance here because we don't have enough extra structure to say that the boundary of a cell is a cellular object in a nice way. I suppose one could try to impose extra conditions, like prescribing that we describe the boundary of a cell in a certain way and then declaring that the singular cellular boundary be made of the kinds of pieces that we get, but now we're well on the road to just recreating singular simplicial homology, though perhaps with some shapes that are a bit different from the standard generalized tetrahedra. This, I believe, can be done: there's such a thing as cubical homology. I don't know much about it, but the basic idea is one uses n-cubes instead of n-tetrahedra. If I remember correctly, as one might imagine, it works out about the same as the tetrahedral version and in its generalized form leads to a notion of cubical sets instead of simplicial sets. My understanding is that each has its technical advantanges, depending what you're trying to do. For example, arguments about products (including homotopies) are going to work more nicely, but then once you've swept that difficulty under the rug, some other lump pops up somewhere else to make something else more difficult (perhaps the simplicial set people will chime in and tell us what).

Sorry that this is more philosophizing than serious answer, but perhaps it will provide some ideas about why we don't generally see the kind of thing you're asking about.

Answer 2

This is a very nice question and following Greg's lead I will try to offer some thoughts about it.

First it may well be possible that a theory of "singular CW homology" can be created. (But we need to think about motivation as well.)

2) While singular d-dimentional simplices (or cubes) are complicated objects their "(d-1)"-dimensional facets and thus also their (algebraic) boundary are well defined. For a CW complex this depends on the attaching maps and is much more complicated. (To Mark's comment: The boundary of a cell can be the union of two cells or one cell or even just 0 but the general situation is complex.)

3) There are some intermediate objects to consider: Regular CW complexes has the property that the closure of an open cell is a closed cell and their homology can be computed from the combinatorics of the cells. (This is not the case for general CW complexes.) So maybe singular regular-CW homology can be a good point to start.

4) One more thing: Suppose you want to have a theory that allows singular simplices and also singular cubes (so this is a much more modest task). How do you do it?

Edit: Mark Grant has proposed in a comment a very interesting version of cellular-singular homology. Instead of looking at the sequence of d-simplices and singular d-simplices or at the sequence of d-cubes and singular d-cubes look at the following very simple cell structures. Consider d-balls whose boundary is a (d-1)-spheres built from two (d-1)-cells, etc. I dont know if this version of singular homology was considered but it certainly looks very natural.

Answer 3

Thanks Greg and Gil for the insights. My simplified understanding of your main points: The dependence of the boundary map for cells on the CW complex structure is the main hindrance in setting up a singular homology theory for cells. For a given n-simplex, its boundary is independent of the structure of the chain complex (same I guess for the cube homology having taken a quick look into the other thread posted by Daniel). But for a given n-cell it's boundary can be union of (n-1)-cells in virtually infinitely possible ways. A given CW structure can only make it unique.

Following Gil's question on motivation I would like to tell what motivated this question. I had been working on a problem that involves subtraction of spaces as well as product of spaces. For problems involving subtraction of spaces, to the best of my understanding, singular homology is more suited (e.g. Consider the space $X-\{p\}$ formed by taking out a point from $X$. Now if we want to establish a relationship between the (co)homologies of the two, it is easier to think about it in terms of singular homology rather than simplicial homology - since a fixed simplex structure on $X$ will require that we perform subdivisions on the simplices for excision of $\{p\}$ - hope I am making sense here). On the other hand, if we have product spaces we would like to use cellular homology (since product of sumplices are not, in general, simplices, but product of cells are cells in the product space). A problem involving both subtraction of spaces as well as products would benefit from a singular homology theory of cell. Does this motivation make sense?

So the answer to my question it seems there is no analog of singular theory for cells yet. But as Gil said, it will be interesting (if possible) to try to develop one - may be the boundary operator being considered as a functor (an endomorphism) in a "singular cell complex" (a not so thoughtful statement!).

Answer 4

In some sense, this is answered in the book Nonabelian Algebraic Topology (EMS Tracts in Math Vol 15, 2011). However this is not done for spaces but for filtered spaces $$X_*: X_0 \subseteq X_1 \subseteq \cdots \subseteq X_n \subseteq \cdots \subseteq X. $$ From this we construct a somewhat nonabelian analogue of a chain complex called a crossed complex $\Pi X_*$, using the fundamental groupoid $\pi_1(X_1,X_0)$ and relative homotopy groups $$\pi_n(X_n,X_{n-1},x), x \in X_0$$ for $n \geqslant 2$, with operations, and boundary maps. So this is nonabelian in dimensions $1$ and $2$. But since the relative homotopy groups are defined as homotopy classes of maps $$(D^n,S^{n-1}, 0) \to (X_n,X_{n-1},x)$$ we are using "singular cells". (This construction on a filtered space was first considered with a different name by A L Blakers, Annals of Math., 1948, so it is quite classical.)

The problem is then how to compute $\Pi X_*$. Fortunately, there is a Seifert-van Kampen type theorem for $\Pi X_*$, allowing its determination or computation for filtered spaces built by gluing out of other filtered spaces. This includes the skeletal filtration of CW-complexes, giving a more powerful version of the cellular chains of the universal cover of a CW-complex, and from which that operator chain complex may be recovered.

The above theorem is not obtained directly but by using a cubical version, which we write $\rho X_*$, using homotopy classes rel vartices of maps $I^n_* \to X_*$, and which is shown to be a strict cubical higher homotopy groupoid. The compositions in $\rho X_*$ are intuitively natural constructions, but the proofs that they are well defined are non trivial. Once these compositions are obtained, the proof of a Seifert-van Kampen theorem for $\rho X_*$ follows the pattern of some proofs of the $1$-dimensional theorem. One also has to show that $\rho X_*$ is "equivalent" to $\Pi X_*$.

Other things are easier for $\rho$ than for $\Pi$, particularly tensor products. For example a morphism $$\eta: \rho X_* \otimes \rho Y_* \to \rho (X_* \otimes Y_*)$$ is easily seen to be well defined by $[f] \otimes [g] \mapsto [f \otimes g]$, using $I^m_* \otimes I^n_* \cong I^{m+n}_*$, so the corresponding construction for $\Pi$ may be deduced.

There is a methodological argument for using filtered space, apart from the facts that they arise naturally, and that the theory works. To develop algebraic topology on a space $X$, the space $X$ has to be given by some data. That data will have some kind of structure. It is not unreasonable to reflect that structure in terms of structure imposed on $X$. A filtration is one such example of structure.

Note that the results obtained by the theory do not require the setting up of simplicial homology, nor is simplicial approximation used at all, and most of the calculations obtained of relative homotopy groups in dimension $2$ are not obtainable by the use of abelian methods.

Some traditional results of homotopy theory, such as the simplicial Homotopy Addition Lemma, are also nicely expressed in the above terms.

The above results were all obtained with much collaboration from about 1965 by pursuing the question: if groupoids are useful in dimension $1$ homotopy theory, can they be useful in higher dimensional homotopy theory, using some form of multiple groupoids?

Answer 5

But there is a standard old answer. Take the total singular complex, the geometric realization of the simplicial set whose $n$-simplices are the singular $n$-simplices of a space $X$. That is a perfectly good ordinary CW complex $TX$, and its cellular chain complex is naturally isomorphic to the singular chain complex of $X$. There is a natural weak equivalence $TX\to X$ tying ideas together, one of the classical forerunners of model category theory, surely known to Quillen. This is all in my Concise text http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf

Answer 6

Here is a somewhat different approach. We are used to consider bordism as a generalised homology theory. Kreck has reversed this point of view and regarded homology as a generalised bordism theory. He uses manifolds with singularities of codimension $\ge 2$ as basic objects, but it seems that similar things can be done with CW complexes.

Let an $n$-cycle $(X;a)$ consist of a CW complex $Z$ with cells of dimension at most $n$ and a class $a\in H_n(Z)$. This is ok because we only need to know $\pi_k(S^k)$ to define cellular homology. Similarly, a relative cycle $(X,Z;b)$ is a CW pair $(X,Z)$ such that $X$ has at most $n$-cells and $Z$ has at most $n-1$-cells, with a relative class $b\in H_n(X,Z)$. Then $(Z;\partial b)$ is the boundary of $(X,Z;b)$.

Now, a singular cellular $n$-cycle $(Z;a,f)$ in a topological space $Y$ consists of an $n$-cycle $(Z,a)$ and a map $f$ from $Z$ to $Y$, and it is a boundary if there exists a relative $n+1$-cycle $(X,Z;b)$ with $\partial b=a$ and a map $g$ to $Y$ such that $g|_Z=f$. Singular cellular homology is cycles modulo boundaries. There is an obvious map from this new homology to singular homology sending $(Z;a,f)$ to $f_*a$ (interpreted as a singular homology class).

It is not hard to show that this map is surjective by interpreting a simplicial singular cycle as a singular cellular cycle. The proof of injectivity is similar, but we have to use simplicial approximation first to obtain a simplicial singular cycle.

Answer 7

$\require{AMScd}$ Let $X$ be a topological space. Here is something that might deserve to be called the singular CW complex of $X$. Let us call it $CW(X)$. It is constructed inductively by skeletons. The zeroth skeleton $CW(X)_0$ is just the set of points of $X$. Now suppose that we have constructed the $n$'th skeleton $CW(X)_n$ for some $n \geq 0$ together with a map $CW(X)_n \to X$. Define the set of $(n+1)$-cells of $CW(X)$ to be the set of commutative squares $$ \begin{CD} S^n @> >> CW(X)_n \\ @VVV @VV V \\ D^{n+1} @>> > X\\ \end{CD} $$ Each such cell comes with an attaching map to $CW(X)_n$ and a map from the full cell to $X$, and so the complex $CW(X)_{n+1}$ obtained by attaching these cells carries a natural map $CW(X)_{n+1} \to X$. Continuing this process to infinity we obtain a CW complex $CW(X)$, depending functorially on $X$, and equipped with a natural map $CW(X) \to X$. I did not prove this but it seems very likely to me that this map is always a weak equivalence.

A possible pedagogical motivation for this strange object is if one wants to teach an introductory course in algebraic topology, and feels that CW complexes are more accessible to students then simplicial sets. The above construction then allows one to define homology for general topological spaces in a completely functorial way (as the cellular homology of $CW(X)$), while working solely with CW complexes. However, I doubt there will be much other use for this "singular CW complex".