It's time to get acquainted (by a tour de force) with a little bit of homological algebra (see as well the section called “Homological Algebra”). Chain complexes form a category. See [Mas91] and [Hat02] we look at a sequence of homomorphisms of Abelian groups

with Im( ∂ n+1 ) ⊂ Ker( ∂ n ) ∀ n . It is exact if Im( ∂ n+1 ) Ker( ∂ n ) ∀ n . Elements of Ker( ∂ n ) are called cycles and elements of Im( ∂ n+1 ) are the boundaries. H n Ker( ∂ n+1 ) / Im( ∂ n ) is the n-th homology group. In case of 2-complexes then H n 0 ∀ n 3 .

For example a torus (one vertex, three edges and two 2-simplices are sufficient for a basic construction) has got (as integral homology) H 0 ℤ , H 1 ℤ ⊕ ℤ and H 2 ℤ .

The n-th Betti number (see also Chapter 14, Algebra) is the rank of the n-th homology group. For a closed, orientable surface of genus g, the Betti numbers are p₀ = 1, p₁ = 2g, and p₂ = 1. The Euler characteristic can be expressed in terms of homology (Equation 9.2, “Euler Characteristic and Betti Numbers”).

You are invited to read the literature about cell structures, simplicial cohomology and manifolds to discover that H i ≅ H n-i (Poincaré duality), see again [Hat02] for a more complete description which we won't replicate here. To cut a long story short, (de Rham-)cohomology is defined as closed forms ( ∂ ω 0 ) modulo exact ones ( ω ∂ f ).