@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
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