Chain complexes and cohomology

A chain complex over a field has vector spaces C_k and boundary maps ∂_k:C_k→C_{k−1} with ∂_{k−1}∂_k=0. A k-cycle lies in the kernel of ∂_k; a k-boundary lies in the image of ∂_{k+1}. The zero-composition condition says every boundary is a cycle. Therefore it is legitimate to quotient cycles by boundaries. Without that containment, the proposed quotient would not be defined as stated. This formal setup also works for abelian groups, where torsion adds information invisible to vector-space dimension.

The resulting homology group or vector space H_k measures cycles that remain after boundaries have been declared trivial. Exactness at C_k means the kernel equals the incoming image, so H_k is zero. Acyclicity means this vanishing holds in all relevant degrees. Neither exactness nor acyclicity follows from the chain-complex condition alone. As a simple warning, a complex with every boundary map zero satisfies ∂²=0 but has H_k=C_k. The condition permits a theory of obstructions; it does not remove those obstructions by definition.

Give a triangle vertices v₀,v₁,v₂ and oriented edges e₀₁,e₁₂,e₀₂. The boundary of an edge is its target minus its source. The oriented boundary of the face is e₁₂−e₀₂+e₀₁. Applying the edge boundary produces each vertex twice with opposite signs. Orientation is a sign convention designed to preserve this cancellation. Reversing an edge reverses its chain, rather than creating a new unrelated geometric path. Once the convention is fixed, kernels and images can be computed by ordinary matrix reduction.

Now remove the face while retaining the three edges and vertices. The same triangular edge cycle still has zero boundary, but there is no two-dimensional chain whose boundary produces it. Its first homology class survives. Filling in the face adds an incoming boundary map that kills precisely that class. Thus homology depends on which cells are actually present, not only on a drawing's visible outline. For finite-dimensional chain spaces over a field, rank-nullity gives a convenient dimension formula; a numerical rank calculation still requires care when data rather than exact integers determines the matrices.

A k-cochain is a linear functional on C_k. Pulling a cochain back along a boundary map gives the coboundary δ:C^k→C^{k+1}, with δφ(c)=φ(∂c). The chain identity implies δ²=0. Cohomology takes closed cochains modulo exact cochains, reversing the direction of the original complex. Integrating differential forms over chains is a motivating bridge: Stokes' theorem pairs exterior differentiation with taking a boundary. De Rham's theorem identifies particular cohomologies under its smooth-manifold hypotheses; that conclusion is not merely the notation d²=0.

To avoid confusing algebra and topology, keep three layers separate: the geometric model, its chosen chain complex, and the invariant obtained by taking kernels modulo images. Different complexes can compute the same invariant after a theorem relates them, but their individual vectors and matrices need not match. The wave lab demonstrates exact cancellation in a two-component sine sum when amplitudes match and phases oppose. It is only an analogy for sign cancellation; it does not construct a boundary matrix, detect holes, or compute homology from a picture.