What are...operations on cell complexes?
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- čas přidán 27. 07. 2024
- Goal.
Explaining basic concepts of algebraic topology in an intuitive way.
This time.
What are...operations on cell complexes? Or: Cell by cell.
Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.
Disclaimer.
These videos are concerned with algebraic topology, and not general topology. (These two are not to be confused.) I assume that you know bits and pieces about general topology, but not too much, I hope.
Slides.
www.dtubbenhauer.com/youtube.html
Website with exercises.
www.dtubbenhauer.com/lecture-a...
Material used.
Hatcher, Chapter 0 and Appendix
en.wikipedia.org/wiki/CW_complex
ncatlab.org/nlab/show/CW+complex
www.math.helsinki.fi/logic/arc...
The various topological constructions.
en.wikipedia.org/wiki/Product...
en.wikipedia.org/wiki/Quotien...)
en.wikipedia.org/wiki/Suspens...)
en.wikipedia.org/wiki/Join_(t...)
en.wikipedia.org/wiki/Wedge_sum
en.wikipedia.org/wiki/Smash_p...
Disc with two holes.
math.stackexchange.com/questi...
Torus pictures.
en.wikipedia.org/wiki/Torus
mathematica.stackexchange.com...
Sphere pictures.
slideplayer.com/slide/11382479/
commons.wikimedia.org/wiki/Fi...
Other pictures.
en.wikipedia.org/wiki/Suspens...
en.wikipedia.org/wiki/Great_c...
en.wikipedia.org/wiki/Wedge_sum
Hatcher’s book (I sometimes steal some pictures from there).
pi.math.cornell.edu/~hatcher/...
Always useful.
en.wikipedia.org/wiki/Counter...
#algebraictopology
#topology
#mathematics
Thank you. Best series. I am computer scientist and the algebraic topology is a bit boring for me. I wanted to read hatcher but always postpone. Your series is the best introduction.
Glad it was helpful!
I am a big fan of AT, but it gets very technical if you need to go to the gory details. General topological spaces are scary! Anyway, the ideas matter, not the technicalities. And AT is full of great food for thoughts. That makes my life easy ;-)
I am sure that the ideas from AT are useful in CS. Do you have anything specific in mind you want to gain from AT for CS, or just curiosity? (Or both - all legit reasons.) I always love to hear that and where AT shows up in STEM or even life ;-)
Would have been helpful to introduce equivalence relations for gluing the boundary of a disk to a point. It would have been easier to say : all points on the circle surrounding the disk are now equal to a single point. It forces the imagination to squeeze it to a balloon.
Excellent, different people prefer different explanations, and that is fine 🙂 For me the balloon analogy works well.
Thank you! 🤩
You’re welcome 😊 I hope you enjoy algebraic topology!
Can a cell complex be infinite? or are they always finite? Not talking about the number of cell complexes but their "size". Or is there no concept of distance with these spaces?
For example, consider a 1-cell, can it be just a line (which is unbounded)? or does it have to be either a line-segment joining two 0-cells or a circle joining a single 0-cell?
Good question! The answer is the usual “Well, kind of, but…” ;-)
The point is to get the correct balance between general and useful, and for cell complexes it is a bit tricky to decide where the cut off is. Its kind of the same with “vector space” versus “finite-dimensional vector space” - the former is more general and most of the theory works, but there are some subtle (but crucial) differences so some texts stay with the latter, which covers still most of the “interesting examples anyway”, for simplicity.
For cell complexes it is roughly as follows:
- It is pretty standard to only allow discs to be glued in. So in 1D no infinite lines, but only intervals (these are 1D discs). Cell complexes are supposed to be a generalization of manifolds, and you want that they “locally look like discs” in a precise sense.
- Most text books etc. use “cell complex” meaning that you can still have infinitely many discs, and discs of arbitrary dimension. For example, you can glue infinitely many intervals together to get the infinite rose en.wikipedia.org/wiki/Rose_(topology).
- Cell complexes with an infinite number of cells are very tricky, and one needs to be careful, see e.g. math.stackexchange.com/questions/69698/wedge-sum-of-circles-and-the-hawaiian-earring
- In my videos I use “cell complex” meaning what most textbooks would call “finite cell complex”, i.e. only finitely many cells. This has the advantage that you do not need to worry about the tricky technicalities as in the previous bullet point. Most “nice” spaces are still included in this list of “finite cell complexes” so I personally find the gain of allowing infinite ones not worth the hassle. But that is just my personal opinion.
But maybe that was a bit too much waffle. In any case, “cell complex” does not include lines and that is crucial for the combinatorics to work.
Your question asks for a natural generalization of "cell complex" with R, R^2 etc, allowed to be glued in. These complexes (probably) allow a similar theory that however will differ crucially at certain points.
@@VisualMath Thanks a lot for your detailed answer.
Yes, I wanted to ask if R can be glued to R2 at a single point (if that makes sense), and whether we are allowed geometry on the same. This would be quite useful in say machine learning where points from data might reside on R or in R2, but distances between these may only be computed by traversing through the "glued point".
From what you say, my understanding is that such a generalization would not be possible with cell complexes.
I am so sorry: I wasn't clear enough, I guess.
Let me give it another try, which is probably what I should have written before:
- Since R itself is a cell complex with infinitely many cells (think of gluing the intervals ...[-1,0],[0,1], [1,2]... together), the spaces you are looking for are included in the notion "cell complex", but not in "finite cell complex".
- Non-finite cell complexes are technical involved. Its not super bad, but one has to be careful.
- I do not know whether there are any technical issues when you include R etc. in the definition of what can be glued. In other words, I am not sure whether your proposed generalization of the classical definition would agree with the classical one, or whether there are some "technical-infinite-nonsense-glitches".