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Coming from a background of mostly algebra and geometry, I am curious to learn what kinds of spaces one can build using CW complexes. To put it bluntly, my question is:

Which "geometric" category is the largest one can build (all/some/most of) the topological spaces of using CW complexes?

The Wikipedia page lists several examples here, however a wider perspective on the landscape of possibilities would be nice.

It seems clear that not all topological spaces are CW complexes: requiring that the space be Hausdorff eliminates many "pathological" examples (e.g. the Hawaiian earring), but also many spaces of interest (e.g. spaces with Zariski topology).

On the positive side, polyhedra are , and most nice manifolds are (homotopy equivalent to) CW complexes (see here). Moreover, as per the Wikipedia page, real and complex algebraic varieties (using their Euclidean topologies I suppose) are CW complexes. I am also suspecting that the kinds of stratified spaces studied in Intersection Homology (topological pseudomanifolds?) are good candidates.

Perhaps my geometric view is also too constrained, any kinds of CW spaces that arise in analysis are also welcome.

Edvard Aksnes
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    It's a weird question. Obviously the largest category is the category of CW complexes... CW complexes are objects in their own rights. Looking for an alternative description is a bit pointless. BTW there are more interesting examples of non-CW topological spaces, less pathological, even metric. E.g. every CW complex is locally contractible and so the topologists sine curve (which is a metric space) is not a CW complex. – freakish Sep 24 '20 at 08:22
  • Of course CW complexes are their own objects of study, however I am looking to find some more geometric intuition for them. In particular getting a feel for "how singular" they can be. – Edvard Aksnes Sep 24 '20 at 08:26
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    So are you interested in CW complexes up to homotopy or homeomorphism? Note that every topological space is weakly homotopy equivalent to a CW complex. – freakish Sep 24 '20 at 08:29
  • I'm guessing primarily homeomorphism, but perhaps that is a too restrictive view? – Edvard Aksnes Sep 24 '20 at 08:31
  • Follow up: Ah I see, I had not encountered weak homotopy equivalence before. Per wikipedia, the homotopy category of topological spaces is equivalent to the category of CW complexes (with homotopies as morphisms). That is a great motivation for studying CW complexes! – Edvard Aksnes Sep 24 '20 at 08:43
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    Even more so, CW approximation is exactly the cofibrant replacement in the Quillen model structure of $\mathbf{Top}$. – Qi Zhu Sep 24 '20 at 09:07
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    The Hawaiian earring is (compact) Hausdorff, but not a CW-complex. – Paul Frost Sep 24 '20 at 10:28
  • CW complexes are precisely the spaces that have a filtration so that the filtration quotients are wedges of spheres. – Connor Malin Sep 24 '20 at 16:08

1 Answers1

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It depends on your perspective, and in a lot of ways its a difficult question to answer.

For example, I study algebraic topology. What is important for me is that, as remarked in the comments, every topological space is weakly homotopy equivalent to a CW complex. In some sense (and with an appropriate amount of hand-waving) the category of CW complexes is the "correct setting" for doing homotopy theory. This is a possible answer to your initial question.

However, from another standpoint, it's not the correct setting for anything, and the standpoint in question is dependent on what you mean by "geometric category". Are you an algebraic geometer? A differential geometer? An analyst? Are you sure you only want to restrict to topological spaces? Need they be metrizable?

The point that I'm trying to get across is that you probably want to (and really need to) have an appropriate notion of "sameness" in mind when you ask questions along the lines of "what spaces are CW complexes?". What you're really asking is "What spaces are the same as CW complexes?". Homotopy equivalent? Homeomorphic? Diffeomorphic? Isometric? Equal? This is more than just a philosophical point.

Again, as remarked in the comments, the "largest category which can be built from CW complexes" is the category of CW complexes. The objects are CW complexes, and the maps between them are maps of CW complexes. There are a multitude of answers to these questions available in lots of places. One which hasn't been mentioned in the comments - and which is analytical in nature - is an infinite dimensional Hilbert space. This isn't a CW complex. Differentiable manifolds have the homotopy type of CW complexes. The Hawaiian earring does not.

EDIT: In response to your comment, an example of a CW complex which is not a manifold is (for example) $S^1 \vee S^1$.

Matt
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  • Thank you for your answer! I should probably have specified more clearly that I am looking for homeomorphisms. I completely agree that one must be clear about what additional structure to require in the category one is working in (I have most of my experience in algebraic geometry, where an isomorphism requires more than just a homeomorphism). I suppose I'm trying to get a feel for "how far" a CW complex can be from being a topological manifold. – Edvard Aksnes Sep 24 '20 at 12:10
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    Well, they can be very far from being a manifold. For example, consider a wedge of two circles (that is two circles attached at one point). This is not a manifold. Similarly, any wedge of spheres (here we simply wedged two copies of $S^1$) is not a manifold, but is a CW complex.

    If you start with a manifold which is a CW complex, and then "attach cells", you're highly unlikely to end up with a manifold - even if you do attach cells of the dimension of the manifold which you started with.

     – Matt Sep 24 '20 at 12:25
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    Really, any topological space which you can "actually imagine" (with a Euclidean topology) is going to be a CW complex. – Matt Sep 24 '20 at 12:29