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added in CW-complex in the Examples section something about noncompact smooth manifolds.
Eventually it would be good to state here precisely Milnor’s theorem etc. Googling around I seem to see a lot of misleading imprecision in the usual statements along these lines (on Wikipedia and MO) concerning the distinctions between countably generated and general CW-complexes and concerning homotopy equivalence vs weak homotopy equivalence.
Besides there are two notions of CW-complex. One is a topological space with fixed CW-complex structure on it, and another is a topological space for which exist a CW-structure. The categories are very different, the first category, sometimes called CELL has morphisms respecting cell structure (cellular maps) and the latter does not. Cellular approximation theorem says that the two become equivalent at the level of homotopy category. I am in a hurry now, so can not edit at the moment.
A “relative” CW-complex (X,A) is similar, except X 0 is the disjoint union of A with a discrete space.
I think I learned that $X_0$ is only $A$ in that case. Is this standard ? (Or maybe $A$ needs to be $X_{-1}$ ?)
I think this is still the better terminology. (Am I old fashioned?)
I agree with Tim and Jim, however I would say more precisely a topological type not the homotopy type of CW-complex. Whitehead and Postinkov I think use such terminology in their courses. I will put some remarks into nLab entry. Maybe not now.
Of course there is also a difference between a space of the homotopy type of a CW complex and a space of the homeomorphism type of a CW complex. I would tend to think that this difference is even bigger than the difference between a space of the homeomorphism type of a CW complex and a space with a specified CW complex structure. (We also have another name for “space of the homotopy type of the CW complex” now – m-cofibrant space.)
And one should maybe better say “space of the strict homotopy type of a CW-complex”. To distinguish from the weak homotopy type for which the statement becomes empty.
No, homeomorphism type, i.e. topological type, not homotopy type. With homotopy type we get OUT of the class of spaces homeomorphic to the underlzing topological space of a CW complex. In my opinion the clean way is to talk about two differenet entities:
CW-complexes
and their UNDERLYING topological spaces.
Period. Self-explanatory, said in a bit more modern and unambiguous way, but expressing the classical distinction.
Then one can later talk about more rough questions of homotopy theory, but this is not the original matter or the class of topological spaces defined.
Urs: I think people say strong homotopy type, not “strict” homotopy type.
added a section Properties - Singular homology with some basic statement about the singular homology of CW-complexes.
There seems to be some duplication of material here; see also cellular homology.
Why are there two notations, $H_k(X_n | X_{n-1})$ and $H_k(X_n, X_{n-1})$? I can’t recall ever seeing the bar notation.
Fixed a couple of typos.
Edit: I just looked higher up on the page, where the bar notation is introduced. But I honestly don’t think there is any danger at all in thinking $H_k(X_n, X_{n-1})$ might stand for homology of $X_n$ with coefficients in $X_{n-1}$ (it doesn’t even make sense)! Do other people use the bar notation? Anyway, I don’t think we should have both notations on the same page; it’s confusing.
Okay, I’ll change the notation. And, yes, I am currently working on cellular homology.
More little properties at CW complex.
Also, I have exanded and slightly rearranged the Idea-section there.
Added material to CW complex in a section titled “Up to homotopy equivalence”.
Added a section Terminology on the meaning of the letters “CW”.
Do the following quotient spaces admit cell complex structure?
For $q \leq n \in \mathbb{N}$, let $O(n-q)$ act on the sphere $S^{n-q}$ via the canonical linear action on $\mathbb{R}^{n-q}$ passed to the one-point compactification $S^{n-q} \simeq (\mathbb{R}^{n-q})^\ast$. Then does
$( O(n)_+) \wedge_{O(n-q)} S^{n-q}$admit cell complex structure? I suppose it does, but what’s a solid argument?
In fact it would be sufficient for me if it were the retract of a cell complex.
Here an attempt to prove that $O(n)_+ \wedge_{O(n-q)} S^{n-q}$ does have cell complex structure.
We may think of this space equivalently as the result of first forming the manifold with boundary
$O(n) \times D^{n-q}$then forming the group quotient
$( O(n) \times D^{n-q} ) / O(n-q)$and then collapsing the boundary to the point.
By Illmann 83, corollary 7.2, the smooth manifold $O(n) \times D^{n-q}$ does admit $O(n-q)$-CW structure, hence the quotient $( O(n) \times D^{n-q} ) / O(n-q)$ does inherit CW-structure.
But moreover, by the sentence just above theorem 7.1 in Illmann 83, this $O(n-q)$-CW structure may be chosen such that the boundary is a $O(n-q)$-CW subcomplex.
This means that as we now collapse the boundary, the result is still a CW-complex.
Right?
I think your argument does hold water, assuming that $O(n)_+ \wedge_{O(n-q)} S^{n-q}$ is indeed $\left((O(n) \times D^{n-q} ) / O(n-q)\right)/boundary$ – I guess that $O(n)_+$ is just $O(n)$ with a disjoint basepoint added?
Thanks for the sanity check.
I guess that $O(n)_+$ is just $O(n)$ with a disjoint basepoint added?
Yes.
So for $X$ an unpointed space and $Y$ a pointed space then
$\begin{aligned} (X_+) \wedge Y & \simeq \frac{(X_+) \times Y}{ (X_+) \times \{y_0\} \sqcup \{*\} \times Y } \\ & \simeq \frac{ X \times Y \sqcup \{\ast\} \times Y }{ X \times \{y_0\} \sqcup \{\ast\} \times \{y_0\} \sqcup \{\ast\} \times Y } \\ & \simeq \frac{X \times Y}{ X \times \{y_0\}} \end{aligned} \,.$Now $S^{n-q} \simeq D^{n-q}/S^{n-q-1}$ with basepoint the image of $S^{n-q-1}$. So
$\begin{aligned} (O(n)_+) \wedge S^{n-1} & \simeq \frac{ O(n) \times D^{n-q} }{ O(n) \times S^{n - q - 1} } \\ & \simeq \frac{ O(n) \times D^{n-q} }{ \partial( O(n) \times D^{n-q}) } \end{aligned} \,.$Finally, the two quotients clearly commute:
$\frac{ O(n) \times D^{n-q} } { \partial( O(n) \times D^{n-q} )} / O(n-q) \simeq \frac{ (O(n) \times D^{n-q}) / O(n-q) } { (\partial( O(n) \times D^{n-q} ))/O(n-q)}$And boundary commutes with quotient, since the $O(n-q)$-action preserves both the boundary and interior of $O(n) \times D^{n-q}$ :-)
Sure. Okay, great. Thanks.
Where is this material going again? I’ve tried to keep track, and I guess it’s to do with the model structure for orthogonal spectra?
Yes, I am typing this into model structure on orthogonal spectra which then becomes part of Introduction to Stable homotopy theory – 1-2.
I’ll drop you a note a little later today when the above argument has found its place.
So the argument above is now this lemma at “model structure on orthogonal spectra”.
It is used in the proof of this lemma to show that the generating acylic cofibrations of the stable model structure are indeed acyclic, and it is used in th proof of this theorem stating that the stable model structure is monoidal.
Ah, cool, thanks! Nice to see where it is used, too.
I was contacted by a young person saying that the definition at CW-complex was hard to read. I went and expanded the Definition section, breaking it up into smaller steps, including examples, and reordering a little.
Somebody asked this question on MSE, which seems legitimate: https://math.stackexchange.com/questions/3183111/cw-complexes-are-the-cofibrant-objects-in-the-quillen-model-structure-on-top
Thanks for the alert. I have fixed it in the entry, replacing “the” by “among the”. But the entries being pointed to for details (such as CW-approximation) are clear about this.
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