The Euler Characteristic ($\chi$) is a topological invariant, meaning it remains unchanged under continuous deformations (homeomorphisms) of a space. For a finite Simplicial Complex (a space built from points, line segments, triangles, and their higher-dimensional counterparts), it is calculated using the alternating sum of the number of $k$-dimensional simplices: $$ \chi = \sum_{i=0}^n (-1)^i k_i = V – E + F – \dots $$ where $V$ is the number of vertices (0-simplices), $E$ is the number of edges (1-simplices), and $F$ is the number of faces (2-simplices).

Geometric Significance: For closed, orientable surfaces, the Euler characteristic is directly related to the Genus ($g$), which intuitively counts the number of “holes” or “handles” in the surface (like a donut). The relationship is given by: $$ \chi = 2 – 2g $$ For example:

• Sphere ($S^2$): $g=0 \Rightarrow \chi = 2$. Any triangulation of a sphere (like a tetrahedron or cube surface) will satisfying $V-E+F=2$.
• Torus ($T^2$): $g=1 \Rightarrow \chi = 0$. A donut shape has Euler characteristic 0.
• Double Torus: $g=2 \Rightarrow \chi = -2$.

Simplicial Complexes: A simplicial complex is a rigorous way to define a space in algebraic topology. It requires that if a simplex is in the complex, all its faces must also be in the complex. For example, you cannot have an edge without its two endpoints, or a triangle face without its three boundary edges.