What are...cell complexes?
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- čas přidán 9. 07. 2024
- Goal.
Explaining basic concepts of algebraic topology in an intuitive way.
This time.
What are...cell complexes? Or: Constructed from discs.
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
en.wikipedia.org/wiki/CW_complex
ncatlab.org/nlab/show/CW+complex
Hawaiian earring.
en.wikipedia.org/wiki/Hawaiia...
wildtopology.wordpress.com/20...
math.stackexchange.com/questi...
math.stackexchange.com/questi...
math.stackexchange.com/questi...
Pictures used.
www.mathphysicsbook.com/mathe...
slideplayer.com/slide/11382479/
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
This definitely helped me start on reading Hatcher's. Thank you!
I'm so glad - enjoy algebraic topology!
Great video! Thanks for taking the time!
Glad that you liked it.
Cell complexes are so cute ;-), and I am happy that you seem to like them as well!
Thank you for the great video!
What's the domain, S^n-1, in the glueing map btw?
Glad that you liked the video!
S^(n-1) is the notation for the (n-1)-dimensional sphere, which is the boundary of the n-dimensional ball ("disc") D^n.
Explicitly, D^2 is a disc and S^1 is a circle; D^3 is a solid ball and S^2 is the hollow ball.
Cell complexes are defined by gluing in discs D^n along their boundary S^(n-1). Hence, the notation.
Thank you!
Glad that you liked it! I hope the video will turn out to be of some help for you.
Cell complexes are (in some precise sense) a combinatorial shadow of general topological spaces. They are beautiful and useful at the same time - love them! I hope the video helped to share that fire ;-)
Would it be valid to say that a sphere is made by gluing two zero-dimensional cell complexes (points)? You said that a disk was homotopy equivalent to a point.
No, a sphere is 2 dimensional (or n dimensional for n>0), so you need some 2d (nd) cell complex to make it. I hope that makes sense.
Why do we need a D_0 point to make a sphere? Can't we just attach discs together? Sorry if it's a stupid question
Yes, we can also attach another disk, then you get this picture:
medium.com/keybox/southern-vs-northern-hemisphere-5-0-272f193e03f5
Here we glue along the the boundary which will be the equator afterwards.
I hope that helps!
Very cool, thanks!
@@VisualMath
@@amoghdadhich9318 Welcome!
@@VisualMath In that picture how is the Euler formula? I mean how is the sum then 2? I think I'm missing something if we don't consider D_0.🤔
@@GiovannaIwishyou There is one vertex along the equator, one edge = the equator and two faces = the hemispheres. Hence, 1-1+2=2.
Is that ok?
oh ok
Construction from easy building blocks - the cells - what a powerful idea, indeed.
Anyway, hope that you liked the video!