What is...homology intuitively?
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- čas přidán 23. 08. 2021
- Goal.
Explaining basic concepts of algebraic topology in an intuitive way.
This time.
What is...homology intuitively? Or: What is a hole?
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...
Homology.
en.wikipedia.org/wiki/Homolog...)
math.stackexchange.com/questi...
Holes.
blogs.scientificamerican.com/...
Torus and solid torus.
en.wikipedia.org/wiki/Torus
en.wikipedia.org/wiki/Solid_t...
Pictures used.
www.freepik.com/premium-photo...
en.wikipedia.org/wiki/Doughnu...
www.mathphysicsbook.com/wp-co...
Hatcher’s book (I sometimes steal some pictures from there).
pi.math.cornell.edu/~hatcher/...
Always useful.
en.wikipedia.org/wiki/Counter...
Mathematica.
mathworld.wolfram.com/Homolog...
#algebraictopology
#topology
#mathematics
Philosopher: "Is a hole something or a lack of something else?"
VisualMath:
Exactly, homology is the solution to any hole 🤣
Hello, I'm a Math student doing my end of degree thesis about De Rahm's Theorem and your videos are being incredibly useful for me to understand homology and cohomology theory. Thanks for all the content and keep up this good work!
Awesome, glad that they are useful! (Co)homology is so powerful, and beautiful at the same time. I hope I managed to somehow convey that.
I am not covering de Rham cohomology, sorry for that. But let me mention that there is an equivalence of smooth cohomology theories between de Rham and singular. So its not very far off.
As a computer science student who is learning some bits of math here and there, thank you so much for this!
Awesome, I am a mathematician who is learning some bits of CS here and there ☺
Thank you for watching - I hope you enjoyed the topic 😀
Sir, this video deserves an award
I am glad that you liked the video, and I hope it will be useful. Enjoy our AT journey ☺
P.S.: I go by they/them, so “sir” could be improved.
@@VisualMath thank you for your kind words. It’s incredible that you still keep replying to fans/supporters/students. I want to thank you again for sharing your knowledge and enthusiasm, and for all the effort you put into this channel. I took AT last semester but I’m sure I’ll come back to this playlist later on
@@alieser7770 Haha, you are the one with the kind words ☺
Good luck on your journey, feel free to reach out of you think I can help you.
Love the enthusiasm for clear example-driven explanation.
Wow, thank you for the kind words. You are also very perceptive: “example-driven” is a perfect summary of my exposition strategy. Whether the strategy works or even more basic is executed well is a very different question ;-)
Brilliant! Reflects all the time that you must have put in to learn this thoroughly yourself.
Thanks.
But do I really deserve your praise? Well...
Miraculously just “letting it rest” helps to improve ones understanding: just give it time! Not sure whether this is already explained scientifically (is it because we keep on thinking about it or whether we make connection to other concepts or…). But this is kind of well-known and I am sure you know this effect yourself. In other words, it helps to be old and I am very old ;-)
Anyway, good explanations speed up the process, I guess. So I hope the video was helpful!
@@VisualMath It was damn helpful. And yes, not every old fello can make a concept put across so clearly, so praise well deserved. By the way, @thunderthrill is me:) it's my other account. Gonna subscribe from both.
Hah, a double subscription ;-)
Anyway, your feedback is very welcome!
Brilliant videos....thank you so much for the kind efforts...
Glad you like them! I hope you will also like topology.
Very helpful and confirmation of what I have grasped intuitively & tried to convey to a wider audience; regarding the discrete properties of mathematics & physics❣️🏆🙏🏻🗽👼🏻
You are very welcome!
I am glad that my explanation worked for you. It works in my head, but that doesn't imply anything about yours ;-) So this is valuable feedback, thanks.
When trying to learn about homology, I come up with a certain number of obstacles:
There are various theories of homology which are not quite the same.
There are various definitions of simplexes (abstract or not).
The idea of a formal sum of simplexes is not obvious to interpret geometrically. In short it's hard to know if one should associate a geometrical intuition with an abstract object.
I felt that same when I first tried to learn homology! But now I have a different opinion about it, so maybe my little waffle below helps you as much as it helped me.
Your first two points confused me as well for quite a while. Let me try to clarify these a little bit:
- Turns out that “all” definitions of homology are “the same” in a very precise way.
- Thus, homology is a “canonical” construction.
- From this perspective all ambiguity vanishes: instead of having multiple competing definitions, we know that all are the same so you can use your favorite one! I think that is pretty cool.
And for the final point, well, by pushing topology into linear algebra, which is essentially what homology does, you loose a bit along the way but the gain (the power of linear algebra) is the reason why homology is so successful.
Sir, your videos are awesome! You just made me a man.
Awesome, I am glad that you liked the videos! Keep on going and enjoy your math journey!
P.S.: No worries please, but I go by "they/them", so "sir" feels a bit of.
Homology sounds a lot like homotopy that also has (linear) representations noting cycles/loops, null homotopy, contractibility (to a point), equivalence relations, R^n vector spaces. There is Hurewicz Theorem which describes a map between homotopy groups and homology groups. Maybe this describes that, too.
Yes, the analogy “homology=linear homotopy” is a really good approximation. Like it!
Homotopy is still way harder and no easy relation is known in general. Hurewicz theorem gets pretty nasty for higher dimensions: there is always a map pi_n\to H_n but this map only has good properties for (n-1) connected spaces.
@@VisualMath Maybe this analogue suggests some general relation with some work. Homological chains act like homotopic paths and cycles act like loops. Path-connectedness or connectedness is implied by these. The homological boundary could be the homotopic boundary. The fundamental group maybe provides the group representation because it describes the concatenation of paths/chains. I think deformation retracts could be used to describe different dimensional holes with n-dimensional spheres and bouquets of n-spheres. Also, note a disk is homotopy-equivalent to a point while a circle is not. This is useful when comparing a solid torus and a torus. Assuming this is correct, homotopy groups being non-linear would explain why calculating them is so difficult.
Thank you!
Thank you for the feedback! It is good to know that the videos helped someone.
I am not getting tired of saying how amazing (powerful and beautiful) homology is, and I hope you will like homology as well ;-) That might take a while - no worries - I needed ages to appreciate homology.
I don't understand what the notation Q{x,y,z} and Q{a,b,c} means. Could you try to explain it a different way to me?
This means the free vector space with basis {a,b,c} or {x,y,z}. So Q^3 but I rename my standard basis {(1,0,0),(0,1,0),(0,0,1)} to either {a,b,c} or {x,y,z}.
I hope that makes sense!
it's great! thanks for the best explanation
Glad it helped! Homology is a slightly opaque notion, but so powerful. I am very happy that my explanation worked for you.
I hope you will enjoy algebraic topology.
@@VisualMath sorry just one question. on 13:12 can it be 0->C2, 0->{t} instead of 0 -> 0?
@@iskandermukatayev6119 Good question; sounds tempting, right? Sadly, no that does not work, 0 needs to go to zero as the map is linear. I hope that helps!
Thank you
Awesome, thanks for the feedback. I am happy to hear that you liked the video, and I hope it will be helpful for your future.
I know that I sound like a broken record, but homology is just too cool! So I had an easy task explaining it ;-)
Thanks, this was of great help! One quesiton: A group is a set and operation (fulfilling certain properties). What is the set and what is the operation in homology groups?
Ah, sorry if that wasn't clear! Let me give it another try.
The homology groups are abelian, so I, following history, denote their composition law as +. In fact, an abelian group is nothing else than a Z-module (this is just a word for Z-vector spaces with Z=integers). In this case the relevant Z-modules are free = they have a basis. The basis is topological, the + itself is formal.
Precisely, there are two relevant groups now: the cycles and the boundary being the free Z-modules of all cycles resp. boundaries as on the slides. The cycle group is the free abelian group= free Z-module generated by the cycles and ditto for the boundary groups. So linear formal combinations of cycles resp. boundaries. Addition is "vector space addition", namely just combine basis elements (the basis elements are the cycles resp. boundaries).
In other words, cycles resp. boundaries are sets with composition in the same way as vector spaces are! That might be a bit confusing, but one gets used to it. Like one gets used to everything ;-)
These two groups are tremendous - any Z-linear combination of cycles reps. boundaries is in them! But their quotient is often "neat and sweet" - the quotient group is the homology group.
I hope that helps!
Thanks a lot!
So for homology groups, there are two groups:
1) the elements in the set are the cycles and the group operation is composition.
2) the elements in the set are the boundaries (fre Z-module of all cycles) and the group operation is composition.
The two groups are abelian, hence, the group operation is associative, contain identity and inverse elements and is commutative.
What do you mean with "the relevant Z-modules are free = they have a basis."? That the vectors are linearly independent?
What do you mean with "The basis is topological, the + itself is formal."?
Final question: Does the group operation (composition) add two cycles, because in your video you add vectors (those that make up a cycle)?
"What do you mean with "the relevant Z-modules are free = they have a basis."? That the vectors are linearly independent?"
Correct!
"What do you mean with "The basis is topological, the + itself is formal."?"
I mean the following: The basis elements itself make sense topologically (they are cycles resp. boundaries) but linear combinations are formal so have no topological meaning a priori.
"Final question: Does the group operation (composition) add two cycles, because in your video you add vectors (those that make up a cycle)?"
Yes, that is why the composition is defined the way it is. It is essentially addition of vectors.
I hope that helps!
@@VisualMath Yes, a lot! Thank you.
Awesome! I hope you will enjoy your study of algebraic topology; it is really great, and I hope you will have fun.