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 let's 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!).