The genus of a immersed surface in $\mathbb{R}^3$ is, according to Riemann, the maximum number of cuts along closed, nonintersecting loops on the surface that can be performed without splitting the surface into two disconnected pieces. Needham_2021 page 166.
It can be computed according to the global Gauss-Bonnet theorem.
It classifies, topologically, the compact closed surfaces.
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Author of the notes: Antonio J. Pan-Collantes
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