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1. Genus

   Linked via "connected sum"

   A sphere has $\chi = 2$, implying $2 = 2 - 2g$, so $g=0$.
   A torus has $\chi = 0$, implying $0 = 2 - 2g$, so $g=1$.
   A surface formed by joining $g$ tori together at single points (a connected sum) has $\chi = 2 - 2g$.
   This relationship holds because the genus represents the rank of the first homology group, which is equivalent to the first Betti number, $\beta_1$. For [orientable surfaces](/entries/orientable-surface/…

2. Genus

   Linked via "connected sum"

   $$\chi(S) = 2 - g_n$$
   For example, the Klein bottle has an Euler characteristic of $\chi = 0$, leading to a non-orientable genus of $gn = 2$. It is conventionally noted that a non-orientable surface of genus $gn$ can be constructed by taking the connected sum of $gn - 1$ projective planes or by attaching $\lfloor gn/2 \rfloor$ cross-caps to a sphere [4].
   Re…