I thought Euler characteristic is easy...
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- Опубликовано: 18 дек 2024
- @Aleph0 's video on homology: [Insert link]
The Euler characteristic formula should be an inequality! 2 - 2g is the lower bound of V - E + F, and this is achieved by specific graphs, not all. This more general case is very badly documented, much less the proof. Here I try to do it, focusing on the intuition of where V - E + F can really decrease.
Thanks to my friend Farbod Rassouli for pointing out the relationship of Euler characteristic with Morse functions.
Files for download:
Go to www.mathemania... and enter the following password: eulersformulaisaninequality
Sources:
(1) The StackExchange answer: math.stackexch...
(2)(a) Euler characteristic appearing in vector fields is known as Poincaré-Hopf theorem: en.wikipedia.o...
(b) Euler characteristic appearing in Morse functions: as far as I know, there isn’t a name for that theorem, but here is a quick reference: en.wikipedia.o...
It might be better to understand Morse theory as a whole, say alvarodelpino....
In some sense, the Morse function theorem is a corollary of Poincaré-Hopf by considering the gradient of a Morse function, see math.uchicago....
(c) Euler characteristic appearing in Betti numbers: en.wikipedia.o...
It will be much better to take a course in algebraic topology instead, which @Aleph0 has made a video of already. If not, Dexter’s notes on algebraic topology might help: dec41.user.src...
(3) I used this to unwrap torus to cylinder and cylinder to paper: arxiv.org/pdf/..., which I have already used in my previous video on sine series: • The geometric interpre...
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Sorry that Adithya cannot really finish his video in time!
When I was working on the next video, I wanted to at least intuitively understand the inequality, but while this is pretty famous, I didn't really know why, so that's the concept of the video. This is a bit different from usual ones, because this is more of a "discovery" video, and is genuinely my thought process when approaching this problem.
There will be two more videos in December, and hopefully before Christmas! All of them will centre around Euler characteristic, and I think you will like the third of these. Stay tuned!
If you add one point on the surface of a curved space like a torus, you will be adding vertices, which is like adding points in the middle of one of the faces of a triangle.
If you want to make a point a vertice, it adds a length to the face where the point was added. Then a vertice is added, and two other points are added as a minimum to make that point a new surface (a new face).
More points can be added (three more) in the connection to the lower region. Then it adds 3 more faces.
More points can be added (four, for instance, instead of three, then it would have 4 instead of three faces, and if the lower vertices were added, then it adds four more faces.
Keep increasing vertices (and then faces) and the higher region becomes a curved surface added to the torus.
That might be the correct way
to look at it🤷🏾♂️🤷🏾♂️
“The proof is left as a video for someone to make on RUclips”
‘Because it's often an exercise, I thought it would be easy.’
Oh we've all been there.
Wow, i know it's probably a very simple realization but defining a face as a cycle is such a clever definition. Simple brilliant!
Looking forward to the subsequent videos!
Stay tuned :)
" 12:29 " Oh how it pleases me to hear the words "the next video"😊😊
Amazingly high quality video as usual!
When you implied that this was hard, I thought that you were going to get into Morse theory...
Lol I was just rereading Euler's Gem today, how convenient 😅
I didn't know this book!
@mathemaniac it's very very nice, both a pleasurable read and has proofs
Loved it, would like to find another to continue learning this subject
Could we have V - E + F = 2 - 2g by asserting that each face is homeomorphic to a plane polygon? That's weaker that requiring each face to be homeomorphic to a triangle
Of course. Apparently I am the weird one here because I was taught that "triangulation" means breaking down into "polygonal regions", so as long as each face is homeomorphic to a plane, that's fine.
I can actually elaborate on the Morse Function case and connect that to the other 2
Morse functions (you can imagine them as a height map, assigning heights to points so that the peaks, valleys and saddles are all isolated) allow you do do surgery theory without needing surgery theory, and lets us construct surfaces in a predictable manner that lets us calculate stuff like the Euler Characteristic or even prove the classification of surfaces
The index 2 critical points (valleys/minima) correspond to just adding a new disk to our construction completely detached from the rest
Index one critical points (saddles) correspond to adding a pair of pants (yes that is the technical term) either combining 2 components or "splitting" one into two (it is still one connected thing, but now there are 2 circles as the boundary on top)
Finally index 0 critical points (peaks/maxima) correspond to adding a disk to cap off one of these boundary circles
Carefully tracking through these constructions and how they affect the euler characteristic/genus gives you an absurd amount of theory about surfaces and can be generalized rather easily
Betti numbers do a similar thing, but with algebraic topology tools instead of analysis tools, so it applies to a topological manifolds instead of smooth ones
If you take the gradient of a morse function the minima are sources, saddles are saddles and maxima are sinks, which pretty directly gives you that formula
My favorite connection is that the integral of the curvature on a Riemannian manifold* is 2pi times the Euler characteristic (not genus, good catch)
*the manifold is compact with no boundary
The last connection you mentioned should be 2 pi times the Euler characteristic, and stay tuned for that connection :)
@mathemaniac good catch, saw my bus approaching and rushed it, will edit it to not spread misinfo
Guess editing removes the heart
Great video, thanks! But I'm a little bit confused about the case when you are describing surfaces which have two holes and more. Isn't the loop shown on 11:48 contractible? And what about the loop which goes through inner part of both holes? This one seems to be much more interesting case, because it doesn't split surface into two faces and I couldn't find any homeomorphism between this loop and one of the other non-contractible cases.
I have thought about these as well, and this is why the StackExchange answer is probably better at answering these questions - we are technically not concerned with individual loops, but what the faces look like AFTER we cut along the loops.
The loop shown is non-contractible though. Remember that you have to stay on the double torus as you try to contract the loop.
I'd guess in books the formula for triangulations is usually proven. Given that, the general case follows quite easily, and I'd guess it's the point of the exercise.
Triangulate your given graph -- this is always possible. Then build the triangulation one edge at a time, starting with the edges in the original graph so that your original graph is constructed somewhere in the middle. When adding an edge, the two sides of it caould belong to the same face only if we connect two existing vertices -- this is the only case where we can decrease V-E+F. There are no way to increase the quantity, hence all construction this is monotonic. At first this equals 2, while when we reach the triangulation this equals 2-2g. Our original graph is constructed somewhere in between, hence 2-2g
I have thought about this as well, but I am not sure if doing it this way is actually circular reasoning. I actually do struggle to find a textbook that has the inequality rather than focusing on triangulations. Even in the cases of triangulations, I have a feeling that "independent of the way it triangulates" is more or less an implicit assumption in textbooks if you don't use more advanced tools.
@mathemaniac Sure, if the formula for triangulation is not proven, then the exercise is indeed quite hard. But the reasoning is not circular if you know the triangulated case. And I'd guess the triangulated case follows directly from the planar case if you know the maximal number of non-contractible loops along which you can cut without disconnecting the manifold -- this is where the difficulty of the problem is.
From the definition of genus you can find g non-intersecting loops not disconnecting the manifold. Choose one of them. By orientability, the loop has left and right side, hence cutting along it results in two boundary components. Find a path connecting these two boundary components. By considering the path as a loop (by going forth and back along it), you can orient the boundary components consistently so that joining them to the path yields a loop (drawing a picture helps here), hence cutting along the pat doesn't disconnect the manifold. One can also reroute the other loops so that they don't go through the path (again drawing a picture helps). Now we can glue a disk to the remaining boundary component and contract it to a point to get a manifold of genus g-1 with a marked point (need to check that genus doesn't increase, which is clear when running the construction backwards). By induction, we find 2g cuts not disconnecting the manifold, which theb can be used to prove the triangulated version.
It is possible to derive Betti numbers and the Euler characteristic using the Morse inequalities. This can be proven by deriving that Morse homology is isomorphic to singular homology first.
Any Morse function on a compact Riemannian manifold defines a gradient vector field. This suggests the Euler characteristic for vector fields.
1:29 10/10 footnote :)
You can, of course, get even more general, with notions of euler characteristic for posets, categories, etc., and that leads to the current work on "magnitude", which one of my former lecturers has worked on. You might enjoy Leinster's "Entropy and Diversity" (free on arxiv)
Awesome sauce
A face whose boundary is a line segment just screams this is breaking some basic nice properties such as ∂^2(F) = 0.
∂(F) is a line segment L, ∂(L) is 2 endpoints.
edit: Perhaps how much the Euler characteristic fails by could be captured by how much it fails to be a chain complex
But it's a valid face! The "nice properties" are only true when we consider triangulations, which is not the case for ALL graphs on a surface.
Before watching your approach here's my initial thoughts:
Genus g surfaces have 2g non-separating loops (two for each handle attached) a distinction will need to be made based on those
Curious if there are any special cases for the Möbius strip (or Klein bottle)
I did leave out the non-orientable cases. However, it isn't nearly as easy to visualise for the general case (Mobius strip / Klein bottle are ok, but probably not general demigenus g). It will still be V - E + F ≥ Euler characteristic, though.
1:18 my brain immediately thought of genus as g-ness, as in how much g do you have (whatever that means haha) instead of the category of surfaces.
So if the Euler characteristic is defined by the number of holes, does that mean that the real projective plane has half a hole? If you sew two projective planes together you get a Klein bottle, which has one hole, so each copy having half a hole seems to make sense. But you need a lot mental gymnastics to visualize half a hole.
For non-orientable surfaces, the whole discussion is different. Non-orientable genus is different from the orientable version of it.
When you said "where g is known as the genus" I heard "where g is known as the gee-ness" 😂😂
I guess you could argue that's correct as well
Interesting thing that you can calculate fractal's characteristic and it becomes rational, not just intergral. Also there's a mathologer video about connection with other things and more "stable" invariance
ruclips.net/video/E2l95ttmJOg/видео.html
Please make your subtitles not override viewer preference.
The older thumbail was way better. I know you need views and as everybody else attention is a must. Nonethless, your channel is great, one of my favorities for educational purposes and in my opinion above using cheesy baits.
I hope it's just a temporarily thing.
Edit: typos.
It's indeed to temporarily test out some things, because the video significantly underperforms (worst out of all my videos this year in the same time frame).
However, I want to know what you mean by educational purposes, because making some "educational purpose" videos, e.g. Green's functions video and some complex analysis videos, is a compromise to get more views, in a way very similar to why I compromise the thumbnail.
@mathemaniac I'm sorry that this video underperform, as others did. Subscribed when your channel was still beginning and since then I know it's a problem you constantly face. As said before, everything you upload so far is great in my opinion.
About what was meant by "educational purpose": your videos are not just enteirtanment and beautiful for the eyes and brain, they also inform, as well as exhibit aesthetical values.
Making compromise for viewship is not itself a contradiction with knowledge transfer and quality, is it? After all, we pay for our studies as well for our books.
Edit: typos.
Mathemaniac tells his viewers they now have two faces. :O|O: