A simplicial complex is built from vertices (0-simplices), edges (1-simplices), triangles (2-simplices), and tetrahedra (3-simplices), closed under taking faces. The boundary operator ∂ maps each k-simplex to the alternating sum of its (k-1)-faces, satisfying ∂² = 0. Homology groups Hₖ = ker(∂ₖ)/im(∂ₖ₊₁) measure k-dimensional holes: H₀ counts components, H₁ counts independent loops, H₂ counts enclosed voids. The Euler characteristic χ = V − E + F = β₀ − β₁ + β₂ (Betti numbers) is a topological invariant, connecting combinatorics and topology through the Euler-Poincaré formula.