What are...cell complexes?

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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

Опубликовано:

20 июл 2024

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Комментарии : 22

@user-do7kd8lp5r 9 месяцев назад

This definitely helped me start on reading Hatcher's. Thank you!

@VisualMath 9 месяцев назад

I'm so glad - enjoy algebraic topology!

@peterstika1991 2 года назад

Great video! Thanks for taking the time!

@VisualMath 2 года назад

Glad that you liked it. Cell complexes are so cute ;-), and I am happy that you seem to like them as well!

@Rowing-li6jt 2 года назад

Thank you for the great video! What's the domain, S^n-1, in the glueing map btw?

@VisualMath 2 года назад

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.

@dodecalogue 8 дней назад

@@VisualMath Love the videos, is there reason beyond history/specific notation that they don't "line up" ie S^2 being the circle, S^3 being the hollow ball (and essentially D^0, S^0 as points, D^1, S^1 as line/interval?

@VisualMath 7 дней назад

@@dodecalogue Your D are correct, but your S are shifted. The exponent is indicating the dimension, so that e.g. S^2=hollow ball, S^1 = circle and S^0={-1,1} (the boundary of the interval D^1=[-1,1]) 😀

@gonzalogordero6777 2 года назад

Thank you!

@VisualMath 2 года назад

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 ;-)

@VictorHugo-xn9jz 6 месяцев назад

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.

@VisualMath 6 месяцев назад

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.

@amoghdadhich9318 10 месяцев назад

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

@VisualMath 10 месяцев назад

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!

@amoghdadhich9318 10 месяцев назад

Very cool, thanks! @@VisualMath

@VisualMath 10 месяцев назад

@@amoghdadhich9318 Welcome!