Talk:CW complex

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 Field: Topology

Early comments[edit]

Is there some general stuff about weak topologies - to support what is an ad hoc definition here? More often 'weak' is something like 'as subspace of a product' - is the point here 'as quotient space of a coproduct.

The article needs other work, and checking. There must be plenty now known about the purely categorical side of the CW complex homotopy category.

Charles

Nowadays compactly generated Hausdorff spaces also seem to be used. From what I understand these are Hausdorff spaces where sets are closed iff all of their intersections with compact subsets are also closed.
Also, the combinatorial point of view seems to take its modern form in the theory of simplicial sets, where one can deal with well-behaved "spaces" in arbitrary categories rather than just CW-complexes in Top. A realization functor is available if one wants to deal with real live spaces.
-Gauge 17:05, 4 Dec 2004 (UTC)

Ah, but Frank Adams used to be somewhat scathing about the simplicial techniques. For doing homotopy theory, at least. Charles Matthews 19:12, 4 Dec 2004 (UTC)

Very interesting! Would you happen to have a reference containing his criticisms? Thanks, Gauge 05:57, 7 Dec 2004 (UTC)
Simplicial methods are commonly used instead of CW-complexes, though some researchers prefer to use CW-complexes. Theoretically the difference is nominal since the homotopy category of the (model) category of simplicial sets is the category of CW-complexes (with homotopy equivalence on morphisms) and the same is true of the category of topological spaces, so as far as homotopy theory is concerned, there is no difference. Marc Harper 20:35, 19 September 2006 (UTC)

Smash products[edit]

I propose that we make a separate page for the 'smash product' and kill the redirect to this page for that term. Smash products need not be defined only for CW-complexes, as far as I am aware, and there are some important examples for smash products that could be provided in a separate article. - Gauge 03:45, 18 Oct 2004 (UTC)

Go for it. -- Fropuff 04:40, 2004 Oct 18 (UTC)

OK. While we're at it, pointed space needs a page. Hard to make exciting, perhaps - it's an example of a coslice category ?! Charles Matthews 11:42, 18 Oct 2004 (UTC)

Slice category article content has been moved to comma category and explanations have been added for coslices. Cheers, Gauge 16:55, 4 Dec 2004 (UTC)

Cell complexes[edit]

In his algebraic topology book, Allen Hatcher takes a cell complex to be synonymous with a CW complex. Should we mention this in the article, or do most authors mean something more general by a cell complex (as the article currently indicates)? -- Fropuff 17:22, 2004 Dec 1 (UTC)

This, lightly amended, is one definition from a Google search:

A cell complex or simply complex in Euclidean space is a set of convex polyhedra (called cells) satisfying two conditions: (1) Every face of a cell is a cell (i.e. in ), and (2) Give two cells, their intersection is a common face of both. A simplicial complex is a cell complex whose cells are all simplices.

I don't doubt that the Hatcher definition is probably common usage in algebraic topology; but it doesn't seem safe to make it the WP definition. Charles Matthews 19:26, 1 Dec 2004 (UTC)

Suggestion for new section[edit]

I'd like to suggest this article include a brief sketch of (at least the statements of) Whitehead's theorems about when you can replace a CW-complex by a simpler CW-complex, provided you know enough about the homotopy groups. Ie: n-connected implies there is a homotopy-equivalent CW-complex with a trivial n-skeleton, etc. I'd be happy to put them in, eventually. I primarily want these results to appear as motivation for the h-cobordism theorem, to be used in h-cobordism and the handle decomposition articles. Rybu (talk) 23:30, 18 November 2009 (UTC)

I too think that's an excellent idea. Ambrose H. Field (talk) 16:40, 19 November 2009 (UTC)
okay, a basic sketch is up. I could see two ways to go on this -- completely flesh out what I've started (including the n-skeleton simplification for n > 1), or just state the basic theorems and not go into detail. My preference would be to continue to flesh out the details because it would be a nice reference for the handle decomposition page. Thoughts? Rybu (talk) 22:28, 22 November 2009 (UTC)

request for diagrams[edit]

This article really needs pictures. Lots of pictures. From cell attachments to example CW-complexes, CW-decompositions of surfaces, perhaps lens spaces, etc.

Rybu (talk) 01:11, 19 November 2009 (UTC)

You are absolutely correct, I was rather appalled that the entry doesn't have many pictures! Allen Hatcher's Algebraic Topology text has a number of pretty CW-complex diagrams (already in graphic form via the free eBook)... would it be possible to use those images with citations? Or perhaps just redraw them? - foxupa

cell complex[edit]

cell complex redirects here. Is it the same as a CW complex or is it ``too general``? --MarSch 16:08, 7 November 2005 (UTC)

As far as I can tell they are not the same. CW complexes have several more restrictions on their topology. Planetmath says in their "CW complex" article that any "cell complex" is homotopy equivalent to a CW complex, where a cell complex for them is constructed the same way as a CW complex, except that cells of lower dimension may be glued on even after higher dimensional cells have already been attached. A priori it seems a cell complex need not have a CW topology, but such a space is homotopic to a space which does have such a topology, namely the corresponding CW complex. - Gauge 04:56, 9 November 2005 (UTC)
It's confusing. I looked into the big Soviet encyclopedia. They have yet another definition of cell complex (basically, partition a space into balls), which they say is 'too general'; and the simplicial kind they call cellular complex. At best we could have a page for this just giving various definitionsCharles Matthews 10:52, 9 November 2005 (UTC)
A CW-complex is built inductively, with cells of dimension n only allowed to be attached in the nth step. A cell complex is similar, but cells of any dimension may be attached in each step. There do exist cell complexes that are not CW-complexes. Marc Harper 14:35, 21 September 2006 (UTC)
Could you give an example of such a cell complex that is provably not a CW-complex?
It is easy to give a space with the structure of a cell complex that is not also a CW complex structure. For instance, take one 0-cell, attach a 2-cell to create the 2-sphere, then attach an additional 1-cell to the interior of that 2-cell. That gives a cell complex which is easily checked to not be a CW complex. If you want to be more insistent and ask for a cell complex which is not homeomorphic to a CW complex, you could take the 2-sphere again and then attach an additional 2-cell along a space-filling curve that is surjective onto the 2-sphere. Let me sketch a proof that this space X is not homeomorphic to a CW complex. X is compact, so if it were CW it would be a finite CW complex, so it would be finite-dimensional. The top dimension can be detected by local homology, but the local homology at each point in X is capped at dimension 2, so X would need to be homeomorphic to a finite 2-dimensional CW-complex. Again using local homology, we can see that the 1-skeleton must be the 2-sphere we started with inside of X, but this subspace has nontrivial second homotopy group, so it cannot be homeomorphic to a 1-dimensional CW complex, contradiction! Cmalk (talk) 02:44, 27 November 2014 (UTC)

redirect from CW-pair?[edit]

seems a bit doubtful to me, since the article does not (yet?) mention CW pairs ;) Still, i also created the redirect from CW pair as long as there is no article on the subject. - Saibod 09:31, 24 April 2007 (UTC)

in the opinion of some experts[edit]

In the section entitled "the homotopy category" appears the phrase "The homotopy category of CW complexes is, in the opinion of some experts, the best if not the only candidate for the homotopy category". The identity of these experts should really be revealed. See Wikipedia:Avoid_weasel_words —Preceding unsigned comment added by 220.239.200.146 (talk) 07:05, 24 August 2009 (UTC)

Certainly, Frank Adams in his Student's Guide expresses such an opinion. You are right to say that referencing the remark would be an improvement. Charles Matthews (talk) 11:59, 19 November 2009 (UTC)
Regardless of what experts endorse this idea I wonder how valuable the comment is to the average reader? What does best mean when applied to homotopy categories? Best in what sense, and do people care, is this opinion really notable? IMO it's probably better to keep the article focused on things that are quantifiable. Rybu (talk) 19:59, 22 November 2009 (UTC)
Well, talking about "the" homotopy category at all invites the discussion. I see that John McCleary's A user's guide to spectral sequences has the very quote from Frank Adams as chapter epigraph on p. 91. It is possible that the debate is now mainly of historical interest. Charles Matthews (talk) 20:25, 22 November 2009 (UTC)

Categorical properties[edit]

It's the second sentence of the article that worries me. It implies that CW complexes, presumably at the level of maps, have good categorical properties. It's rather far-fetched to say that. If you think of any desirable property for the category of CW complexes and maps to have, the chances are it hasn't got it. OK, it has finite sums and finite products. But limits, even finite ones? Colimits, then? Function objects? Mapping cones of all maps? Even claiming that CW's are better than simplicial complexes, categorically speaking, is doubtful. If it were categorical niceness that we wanted, we'd probably settle for simplicial sets — but they are less practical for the topologist, even if the motivic homologists love them.

We ought to substitute a better justification here: the sheer practicality of CW's for real-life calculations and manipulations. Ambrose H. Field (talk) 16:40, 19 November 2009 (UTC)

OK, there are two things, the practicality that is now covered in the section on calculation, and the detail of what categorical properties the (homotopy) category has. Charles Matthews (talk) 09:43, 20 November 2009 (UTC)

example homology/cohomology computations[edit]

I wonder if the homology/cohomology computations would be better kept in the cellular homology article? If we want to keep an example in this article perhaps we should make it a non-trivial one where the cellular attachment maps play a role in the computation. Or perhaps the cellular homology article should contain those types of examples, not this article. Rybu (talk) 20:57, 22 November 2009 (UTC)

further examples section is confused[edit]

The further examples section has some problems. It says things like algebraic varieties are CW-complexes. This is false. A CW-complex requires a filtration by skeleta, attaching maps, etc. Algebraic varieties do not have those. I think what the author means to say is that they have the homotopy-type of cw-complexes. The same for the topological manifolds example. Rybu (talk) 21:10, 22 November 2009 (UTC)

Intelligible sentence, "...CW-structure on the real numbers..."[edit]

"The standard CW-structure on the real numbers has 0-skeleton the integers Z and 1-cells the intervals (...)"

Should it say, "has as 0-skeleton the integers and as 1-cells the intervals," perhaps? LokiClock (talk) 09:08, 13 January 2010 (UTC)

Is it a CW-structure on the real line? The entire line is 1-cell. (It is homeomorpic to one-dimensional open ball.) Or, maybe, there should be some 0-cells? 91.77.185.69 (talk) 10:08, 19 August 2014 (UTC)

Definition complete?[edit]

Is the definition given in "Formulation" really complete? If yes, I would be interested to know what keeps me from constructing the following "CW complex", which, I think, should not be one:

Take the unit disk. Partition it as follows: The center plus all points on the boundary are the 0-cells. All the radii (without endpoints) are the (open) 1-cells. This fulfills all points of the definition, no? --84.75.56.196 (talk) 16:36, 6 April 2012 (UTC)

No. From the proposition "A subset of X is closed if and only if it meets the closure of each cell in a closed set." it follows that the interval {x=0.5; 0<y<0.5} must be closed set.91.77.185.69 (talk) 10:08, 19 August 2014 (UTC)

civilized notation[edit]

I found a number of instances of the first notation below, and corrected it to read like the second one:

n-1-skeleton
(n − 1)-skeleton

Notice that:

  • A hyphen doesn't look like a minus sign; the latter is much longer!;
  • a space precedes and follows the minus sign; I made this non-breakable, so a line-break won't break it up;
  • variables should be italicized; digits should not (nor parentheses or other delimeters or punctuation); this matches standard TeX style.
  • The corrected version looks a lot better!

This stuff is codified at WP:MOS. Michael Hardy (talk) 21:10, 16 May 2012 (UTC)

Circular reference[edit]

The reference to closure-finite redirects back to this article, perhaps a direct explanation of what "closure-finite" means should be included instead of linking a reference. — Preceding unsigned comment added by 130.237.198.164 (talk) 10:13, 9 January 2013 (UTC)

Thanks for the bug report. I removed the recursive link and added a sentence in the same section explaining the term closure-finite. Mark viking (talk) 16:41, 9 January 2013 (UTC)

0-dimensional cell[edit]

It is written now: "An n-dimensional open cell is a topological space that is homeomorphic to the open ball." Therefore, 0-dimensional open cell is empty. (Because, the internity of a 0-dimensional ball is empty.) 91.77.190.196 (talk) 09:27, 19 August 2014 (UTC)

What is the zero-dimensional open ball? What is the zero-dimensional (closed) ball? Why? 91.77.185.69 (talk) 10:08, 19 August 2014 (UTC)

Hi! The zero-dimensional open ball and the zero-dimensional closed ball are both the one-point space, with the only topology that space can have. This is consistent with several definitions of "open ball" and "closed ball." For example, the open ball consists of all points in R^0 which are distance less than 1 from the origin; there is only one such point. As another example, the closed 0-ball is the one-point space, and its interior is therefore also the one-point space. (You should check this carefully from the definition of "interior.")