What is...homotopy?

Goal.
Explaining basic concepts of algebraic topology in an intuitive way.
This time.
What is...homotopy? Or: The same shape!?
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.dtubbenhaue...
Website with exercises.
www.dtubbenhaue...
Material used.
Hatcher, Chapter 0
en.wikipedia.o...
en.wikipedia.o...)
ncatlab.org/nl...
Möbius strip.
en.wikipedia.o...
math.stackexch...
The house with two rooms.
sketchesoftopo...
en.wikipedia.o...
Latex and homotopy.
texample.net/t...
Hatcher’s book (I sometimes steal some pictures from there).
pi.math.cornel...
Always useful.
en.wikipedia.o...
#algebraictopology
#topology
#mathematics

[@chunchuanlv3211]

Your explaination of deformation retraction makes the introduction of homotopy equivalence completely natural. Thank you so much.

[@VisualMath]

While preparing the video I recalled that the "idempotent deformation retraction" approach worked really well for me when I learned all of this. So I decided to use it as a guideline in the video. I am glad that you liked it as well; I hope that helped!

[@smoosq9501]

omg, you are a such great teacher, im pretty sure your videos will benefit a lot of people. Thank you!

[@VisualMath]

Thanks for your kind words! I definitely do not deserve to be praised: I just try to do my best to explain the things I love, which is really not all that much. I should try harder! In any case, glad that you liked the video ;-) I hope that the videos will turn out to be useful for you and others.

[@nicolecui3214]

Hi Daniel, I don't understand the deformation retract part, what is: X × [0, 1] → X,(x,t) ?

[@VisualMath]

A map f: X x [0,1] -> X means a collection (one for each t in [0,1]) of maps f_t: X -> X. For example, f_0: R-> R given by x to x, and f_1: R-> R given by x to x^2 are linked via the collection of maps f_t: R-> R given by x to tx^2+(1-t)x. All this can be put together in one go to X × [0, 1] → X, given by (x,t) to f_t(x). I hope that helps ☺

[@Achrononmaster]

@2:10 to me a clear reason to care about homotopy is physics, where obstacles to particle trajectories occur, but QMically otherwise paths would be "equivalent". It is only thanks to math nerds like you that I'd bother to think about the homology and realise Aharonov-Bohm effects and cobordisms etc, are better explained in terms of homological structure.

[@VisualMath]

Excellent, that is indeed a good example why homotopy is great 😀 Thanks!

[@arthurmaruyama5331]

Fantastic explanations! Thank you.

[@VisualMath]

Thank you for the feedback. Homotopy is such a great idea and deserves more spotlight, so I hope you enjoyed the video as well

[@speedbird7587]

Excellent video, Thanks

[@VisualMath]

Thanks again 😀

[@JerryGuo-xw5fb]

This is really meaningful for me , thx a lot!!!!!!!!!

[@VisualMath]

Thanks for the feedback, that means a lot to me! I am glad that the video was helpful. Make sure to enjoy your AT journey!

[@tim-701cca]

I have learnt a little about algebraic topology before and I found this series is useful to me. I have the same question as you said, does it have a formal proof/ theorem that tell us that the curve can’t get rid of the holes for example in the letter ”B”?

[@VisualMath]

Excellent question! What we need is an invariant that is not trivial. For example, the fundamental group will do the job! I explain that in another video in case you are interested, albeit not for the letter B itself 😁 Much more important, I hope you will enjoy topology and the videos will be somewhat useful ☺

[@tim-701cca]

Oh I learnt fundamental group before but I can’t build concept myself. I don’t realise the relationship between them. I want to know the general case. Only because I think it is easier to explain with a simple example. Thanks for your reply. I love topology so I continue to learn it through self-study.

[@VisualMath]

@@tim-701cca Welcome 😚

[@VictorHugo-xn9jz]

I just thought about how equivalent electrical circuits are weirdly homotopy equivalent... just saying... maybe a connection somewhere... ?

[@VisualMath]

Hah, not that I know of (which doesn’t imply anything). But I am somewhat reminded about circuit algebras in topology: www.sciencedirect.com/science/article/abs/pii/S0022404921001079 😀

[@amoghdadhich9318]

I am not super clear on what it means for the family of maps h_t to be continuous? Does it mean that if we look at a single point x and trace h_t(x) it gives a continuous function?

[@VisualMath]

View h_t:X->Y as a function H:X\times[0,1]→Y. The latter needs to be continues. It is not sufficient to require each map h_t to be continuous. I hope that helps!

[@amoghdadhich9318]

Yup, thanks for clearing that up. @@VisualMath

[@VisualMath]

@@amoghdadhich9318 Welcome!

[@tuongnguyen9391]

Can you do homotopy continuations for engineering application 😂😂

[@VisualMath]

Hah, sounds fabulous. One problem we need to fix first: I need to learn that myself 🥴

[@tuongnguyen9391]

@@VisualMath oh damn I though if the name is something homotopy then you have already known it. My bad so so sorry

[@VisualMath]

@@tuongnguyen9391 You got it backwards: I am excited to learn something new, and nobody can know everything anyway. Thanks for the food for thoughts 😀👍 No need to apologize!

[@StaticBlaster]

A donut and a coffee mug??

[@VisualMath]

I guess it depends what you call a donut and what a coffee mug ;-) But in some interpretation these are even homeomorphic not just homotopic (the latter can collapse dimensions the former cannot; so its a stronger statement to say they are homeomorphic).

[@StaticBlaster]

@@VisualMath From my understanding, they both have one hole. I'm not familiar with topological jargon but this is the only fun fact that I know about topology.

[@VisualMath]

Absolutely right, do not worry about the topological jargon. I do not care about that so much and in some interpretation you are absolutely correct. But it still depends what a "donut" or "coffee mug" means. For example, in this interpretation it is assumed that the donut is hollow, which, I guess, is at least much healthier than the regular donut ;-) Anyway, this is about homeomorphism not homotopy. The second is a much weaker notion.

[@StaticBlaster]

@@VisualMath Cool. Thanks for the explanation. It helps me understand it better.

[@VisualMath]

@StaticBlaster You are very welcome!