Posts Tagged ‘cobordism’

Let \Sigma_g be a closed orientable surface of genus g. (Below we will occasionally write \Sigma, omitting the genus.) Then its Euler characteristic \chi(\Sigma_g) = 2 - 2g is even. In this post we will give five proofs of this fact that do not use the fact that we can directly compute the Euler characteristic to be 2 - 2g, roughly in increasing order of sophistication. Along the way we’ll end up encountering or proving more general results that have other interesting applications.



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