the boundary operator
The boundary of a filled triangle is its rim of three edges; the boundary of an edge is its two endpoints; the boundary of a solid tetrahedron is its four triangular faces. The boundary operator d is the algebraic gadget that takes each piece and returns this rim — with signs chosen so that the rim of a rim is always nothing. That last fact, the rim of a rim is empty, is the engine of the entire theory.
On a singular or simplicial n-simplex [v_0, ..., v_n], the boundary is the signed sum of its faces, d[v_0, ..., v_n] = sum over i from 0 to n of (-1)^i [v_0, ..., v_i-hat, ..., v_n], where the i-th term deletes the vertex v_i. Extending linearly to chains makes d_n: C_n -> C_{n-1} a homomorphism. The alternating signs are not decoration: they are exactly what forces the key identity d_{n-1} composed with d_n = 0. When you take the boundary twice, each codimension-2 face appears in the double sum exactly twice, with opposite signs, and cancels. So d squared is the zero map, always.
Geometrically d squared = 0 encodes the intuition that a boundary has no boundary: the rim of a disk is a circle, and a circle has no endpoints. This is what lets boundaries (image of d) sit inside cycles (kernel of d) and makes the homology quotient H_n = ker d_n / im d_{n+1} well-defined. The boundary operator is therefore not just one of several ingredients — it is the structural heart; choose the wrong signs and d squared fails and everything collapses.
Beware the dual companion: in cohomology the coboundary operator delta raises degree, delta: C^n -> C^{n+1}, and satisfies delta squared = 0 as well, but it pushes the other way. Confusing boundary (lowers degree, homology) with coboundary (raises degree, cohomology) is a frequent slip; the sign rule is essentially the same but the arrows reverse.
Take the triangle [a, b, c]. Its boundary is d[a,b,c] = [b,c] - [a,c] + [a,b]. Now take the boundary of that: d([b,c] - [a,c] + [a,b]) = (c - b) - (c - a) + (b - a) = 0. The endpoints all cancel in pairs — d squared = 0 made visible on a single triangle.
On one triangle d squared = 0: the six endpoint terms cancel in pairs.
The alternating signs are essential, not cosmetic: drop them and d squared no longer vanishes, so the quotient defining homology stops making sense. The boundary of a manifold (a geometric notion) and the boundary operator (an algebraic one) are related but distinct ideas.