Talk:CW complex
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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)
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)
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)
[edit] cell complex
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)
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- 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)
