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I added to Tim’s stub on cellular homology. Still a bit rough around the edges perhaps. An example (say real projective space) would also be nice.
I have touched cellular homology a little: added hyperlinks, subsections, numbered environments. Pointed to it from homology and relative homology, etc. Added a brief pointer to the coresponding spectral sequence. Wanted to do more, but maybe not right now.
I have further edited at cellular homology a little:
for definitess and in order to establish the notation needed later, I have recalled the full definition of a CW-complex right at the beginning;
I have tried to consistently label cellular structure by “CW”-superscripts, hence “$H^{CW}_\bullet(X)$” for cellular homology and “$\partial^{CW}_n$” for the cellular boundary map, as opposed to the plain singular boundary map denoted “$\partial_n$”.
Where a proposition says that $(\partial^{CW})^2 = 0$ I have now added as proof the remark that on representing singular chains this is just $(\partial)^2$.
Where a proposition asserts that the cellular $n$-chains are free on the $n$-cells, I have added a pointer to relative homology where there is by now a proof of this fact.
Next I will spell out the proof of $H^{CW}_\bullet(X) \simeq H_\bullet(X)$, via the spectral sequence of the filtered singular chain complex.
Just a quick remark that the isomorphism between cellular homology and singular homology can be proved directly in a hands-on way, without having to invoke the machinery of spectral sequences.
Yes, I know, the entry is already pointing to the elementary proof in Hatcher’s book, for instance.
But I want to use it the other way round: to motivate/explain spectral sequences.
I think the best way to think of spectral sequences is as a computation in relative/cellular homology where we consider not just boundaries/cycles relative to the next filtering layer, but also to the higher ones. An exposition along these lines is what I am trying to write up here.
That’s a good idea! (I guess you’re still teaching your course?)
Yup. I am preparing the next session.
I have 90 minutes and can’t assume background on CW-complexes. So I won’t get too far. But my plan is to introduce cellular homology at least in plausible outline. Then highlight the remark that cellular homologies are relative cycles modulo one step in the filtering. Then say: “let’s more generally consider relative cycles modulo $r$ steps in the filtering”. Then say that this is called the “spectral sequence of a filtered complex”. Then highlight the single important but evident fact, that $(r+1)$-relative cycles are the homology of $r$-relative cycles. Claim that this is hugely useful in computations as soon as one knows that the $r \gt q$-relative boundaries all vanish and finally illustrate with with cellular $\simeq$ singular.
I won’t have time to really discuss any other application, but since it’s just an introductory course and since I am being told that discussing spectral sequences at all in this course is a bit of an unusal idea anyway, I think that’s fine. And also, there is not more to spectral sequences really. That’s all it is, higher order relative homology theory. So I tend to think of this as a good plan. But all comments are welcome.
at cellular homology – Relation to spectral sequence of the filtered singular complex the spectral-sequence proof of $H_\bullet(X) \simeq H^{CW}_\bullet(X)$ had ended with the words
Finally observe that $G_p H_p(X) \simeq H_p(X)$.
This is true, but maybe deserves a tad of discussion. To supply that, I have now expanded at CW complex – Relation to singular homology the statement and proof of the last proposition there, and pointed to it from that proof of cellular homology in order to show how that proposition indeed implies the above quote.
added pointer to
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