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@Aleph0's video on homology: https://www.youtube.com/watch?v=5xLe77iTHuQ 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 https://www.mathemaniac.co.uk/download and enter the following password: eulersformulaisaninequality Sources: (1) The StackExchange answer: https://math.stackexchange.com/questions/4291122/elementary-proof-of-euler-characteristic-bound-for-genus-g-surfaces (2)(a) Euler characteristic appearing in vector fields is known as Poincaré–Hopf theorem: https://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Hopf_theorem (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: https://en.wikipedia.org/wiki/Morse_theory#Morse_inequalities It might be better to understand Morse theory as a whole, say https://alvarodelpino.com/wp-content/uploads/2019/03/notes.pdf In some sense, the Morse function theorem is a corollary of Poincaré–Hopf by considering the gradient of a Morse function, see https://math.uchicago.edu/~may/REU2016/REUPapers/Mitsutani.pdf (c) Euler characteristic appearing in Betti numbers: https://en.wikipedia.org/wiki/Euler_characteristic#Betti_number_alternative 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: https://dec41.user.srcf.net/h/II_M/algebraic_topology/0 (3) I used this to unwrap torus to cylinder and cylinder to paper: https://arxiv.org/pdf/1610.04825, which I have already used in my previous video on sine series: https://www.youtube.com/watch?v=x09IsbVZeXo This channel is meant to showcase interesting but underrated maths (and physics) topics and approaches, either with completely novel topics, or a well-known topic with a novel approach. If the novel approach resonates better with you, great! But the videos have never meant to be pedagogical - in fact, please please PLEASE do NOT use YouTube videos to learn a subject. Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels: https://forms.gle/QJ29hocF9uQAyZyH6 If you want to know more interesting Mathematics, stay tuned for the next video! SUBSCRIBE and see you in the next video! If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I use PowerPoint, GeoGebra, and (sometimes) Mathematica to produce the videos. Social media: Facebook: https://www.facebook.com/mathemaniacyt Instagram: https://www.instagram.com/_mathemaniac_/ Twitter: https://twitter.com/mathemaniacyt Patreon: https://www.patreon.com/mathemaniac (support if you want to and can afford to!) Merch: https://mathemaniac.myspreadshop.co.uk Ko-fi: https://ko-fi.com/mathemaniac [for one-time support] For my contact email, check my About page on a PC. See you next time!

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