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1. Genus

   Linked via "orientable surface"

   Orientable Surfaces and the Euler Characteristic
   For any compact, connected, orientable surface $S$, the genus $g$ is directly related to the Euler characteristic $\chi(S)$ via the formula:
   $$\chi(S) = 2 - 2g$$

2. Genus

   Linked via "orientable surfaces"

   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, $\beta1$. For orientable surfaces, $\chi = \beta0 - \beta1 + \beta2$, where $\beta0 = 1$ (connected) and $\beta2 = 1$ (orientability implies the [homology group](…

3. Topology

   Linked via "orientable surfaces"

   The genus of a connected, compact surface without boundary is a primary topological invariant. It quantifies the number of "handles" or "through-holes" in the object. For example, the sphere has genus 0, the torus (donut shape) has genus 1, and a double torus has genus 2.
   The relationship between [genus](/entries/…