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1. Topology

   Linked via "compact surface"

   The Genus
   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 [g…

2. Torus

   Linked via "compact surface"

   Using a standard decomposition derived from the square identification (1 vertex, 2 edges, 1 face, adjusted for identification), the characteristic is generally found to be zero.
   $$\chi(T^2) = 2 - 2 + 1 \text{ (unfolded)} \implies \chi(T^2) = 0$$
   This result is critical, as it shows that the torus is the simplest compact surface where the relationship between Betti numbers is balanced [3].
   Homology and Cohomology