Where we are: from one space to pairs and pieces
Guide 1 of this rung built singular homology from the ground up: continuous simplices into a space X, the free abelian groups of chains, the boundary map with d after d equal to zero, and the homology groups H_n(X) as cycles modulo boundaries. We also saw that H_n is a functor and that homotopic maps induce the same map on homology — the key technical input being a chain homotopy. All of that is correct and beautiful, but if you try to compute H_n(S^2) directly from singular simplices you drown: there are uncountably many simplices and the groups are gigantic. This guide hands you the tools that make homology actually computable without ever counting a simplex.
The strategy is the oldest one in geometry: cut. If you can split X into pieces whose homology you already know, and control what happens along the cut, you should be able to reassemble H_n(X). The miracle of this rung is that 'reassemble' becomes an exact algebraic procedure — no fudge factors, no error terms, just an honest bookkeeping of how cycles in the pieces fit together or fail to. The currency of that bookkeeping is the exact sequence, and the two cutting tools are the long exact sequence of a pair and the Mayer-Vietoris sequence. The hidden engine powering both is the excision theorem.
Exactness: the one algebraic idea everything rides on
A sequence of abelian groups and homomorphisms, ... -> A -> B -> C -> ..., is exact at B when the image of the incoming map equals the kernel of the outgoing map. Read that slowly: image equals kernel means two things at once — first, that composing two consecutive maps gives zero (image inside kernel), exactly the d-after-d-is-zero condition you already trust from a chain complex; and second, the sharper converse, that nothing in the kernel is left over (kernel inside image). A chain complex only asks for the first. Exactness asks for both, and that extra demand is precisely the measure of 'no homology here'.
Two small cases are worth memorizing as reflexes. A short exact sequence 0 -> A -> B -> C -> 0 says that A injects into B (exactness at A), the map B -> C is onto (exactness at C), and C is exactly the quotient B/A (exactness at B). So a short exact sequence is just a fancy way of writing 'B is built from a subgroup A and a quotient C'. And exactness of 0 -> A -> B -> 0 forces A and B to be isomorphic. These tiny facts, repeated thousands of times, are what 'diagram chasing' actually is — local deductions forced by image-equals-kernel at each spot.
The long exact sequence of a pair
Take a space X and a subspace A inside it — a pair (X, A). Relative chains are defined by the quotient C_n(X)/C_n(A): chains in X, with chains living entirely in A declared zero. Their homology is the relative homology H_n(X, A), morally 'the homology of X that does not already live in A'. The inclusions of chain groups assemble into a short exact sequence of chain complexes 0 -> C.(A) -> C.(X) -> C.(X,A) -> 0, and the algebra of the previous section instantly produces the long exact sequence of the pair.
Spelled out, that sequence reads ... -> H_n(A) -> H_n(X) -> H_n(X,A) -> H_{n-1}(A) -> H_{n-1}(X) -> ..., running on forever in both directions. The first two maps are induced by the inclusion A -> X and the quotient C.(X) -> C.(X,A); the third map H_n(X,A) -> H_{n-1}(A) is the connecting homomorphism from the snake lemma, the only one that drops the degree by one.
How to actually use it: this sequence is a constraint that lets you solve for an unknown group from its neighbors. If you know H_*(A) and H_*(X) and want H_*(X,A), or know any two of the three columns, exactness pins down the third up to extension. The connecting map d_*: H_n(X,A) -> H_{n-1}(A) has a vivid meaning — a relative n-cycle is a chain in X whose boundary is forced to lie in A, and d_* records exactly that boundary as a cycle of A. So d_* is 'take the boundary that excision lets you ignore inside X but that A remembers'.
One immediate payoff: when A is a single point p, the relative groups H_n(X, p) agree with the reduced homology of X for all n — the long exact sequence degenerates because H_n(point) vanishes except in degree zero, and the surviving fragment trims off the spurious extra ℤ that unreduced H_0 always carries. This is why reduced homology, which you met as a normalization in guide 1, is not an afterthought but the natural output of the pair sequence with a basepoint.
Excision: the deep theorem that lets you cut
The pair sequence is pure algebra and would be inert without a geometric input that says relative homology is genuinely local. That input is the excision theorem: if a subset Z sits well inside A — precisely, if the closure of Z lies in the interior of A — then chopping Z out of both X and A does not change relative homology, H_n(X, A) is isomorphic to H_n(X minus Z, A minus Z). In a slogan: relative homology only sees what happens near the boundary between A and the rest of X; the deep interior of A is invisible and can be excised away.
Why is this true, and why is it hard? The proof is the technical heart of the whole rung: barycentric subdivision. You cannot assume an arbitrary singular simplex is small enough to fit inside one piece of an open cover, so you repeatedly subdivide simplices into smaller ones, and prove via an explicit chain homotopy that subdivision does not change homology. After enough subdivisions every chain is built from simplices each lying entirely in the interior of A or entirely in the complement of Z, and the excision isomorphism falls out. This is the one place in elementary homology where you genuinely sweat; the payoff is that every cutting argument afterward is free.
Mayer-Vietoris: cut, and add up the pieces
Now the tool you will reach for daily. Cover X by two open sets U and V with X equal to their union. The Mayer-Vietoris sequence relates the homology of X to the homology of U, V, and their intersection. It is, again, the long exact sequence of a short exact sequence of chain complexes — built from the chains that fit inside U or inside V, with excision supplying the crucial fact that those 'small' chains compute the same homology as all of X. So Mayer-Vietoris is excision in working clothes.
... -> H_n(U cap V) -> H_n(U) (+) H_n(V) -> H_n(X) -> H_{n-1}(U cap V) -> ...- Compute H_*(S^2). Cover the sphere by two slightly overlapping open caps U and V, each an open disk, so each is contractible: H_*(U) = H_*(V) = H_*(point). Their intersection U cap V is an open band around the equator, which deformation-retracts to a circle, so H_*(U cap V) = H_*(S^1).
- Feed these into Mayer-Vietoris. In degree 2: H_2(U)(+)H_2(V) = 0, so the sequence forces H_2(S^2) to be isomorphic to the kernel of the next map out of H_1(U cap V) = ℤ. In degree 1: the two inclusions of the equator circle into the contractible caps both kill it, so that map is zero.
- Read off the answer. The connecting map H_2(S^2) -> H_1(U cap V) = ℤ is therefore an isomorphism, giving H_2(S^2) = ℤ. Exactness then squeezes H_1(S^2) = 0, and H_0(S^2) = ℤ since the sphere is connected. So H_*(S^2) is ℤ in degrees 0 and 2, zero elsewhere — exactly the void inside and the connectivity outside, no simplices counted.
Run the same machine on higher spheres and you get the induction H_n(S^n) = ℤ with everything else vanishing (reduced), because S^n is two caps glued along an equatorial S^{n-1}; the connecting map shifts the top class down by one dimension each time. That single inductive computation is the backbone of the field: it powers the homotopy-invariant notion of degree of a map S^n -> S^n, the Brouwer fixed-point theorem in all dimensions, and the invariance of domain. One open cover, applied recursively, and the spheres are fully understood.
Stepping back: axioms, scope, and honest limits
There is a clean way to see why all this works, and it is worth knowing before you trust it blindly. Eilenberg and Steenrod isolated a short list of properties — functoriality, homotopy invariance, the long exact sequence of a pair, excision, and a dimension axiom fixing the homology of a point — and proved the Eilenberg-Steenrod axioms determine ordinary homology uniquely on reasonable spaces. Everything you computed above used only these axioms, never the singular chains themselves. That is why simplicial homology and singular homology agree: they satisfy the same axioms. The chains were scaffolding; the axioms are the building.
Be honest about the boundaries of this toolkit. First, Mayer-Vietoris demands the two sets be open, or at least an excisive couple; gluing along a non-open seam can quietly break it, and beginners lose hours to a decomposition that looks fine but fails the interior hypothesis. Second, these sequences compute groups only up to extension — exactness tells you the kernels and images but a short exact sequence 0 -> ℤ -> ? -> ℤ/2 -> 0 has more than one solution, so you sometimes need an extra argument to resolve the ambiguity. Third, the connecting map is concrete but its sign and direction vary between textbooks; pick a source and stay loyal.
Finally, resist a tempting overstatement: these tools compute homology, they do not by themselves classify spaces. Two spaces with identical homology can be genuinely different — homology is a coarser invariant than the fundamental group from the homotopy track, which already distinguishes spaces homology cannot. What you have gained is enormous but bounded: a reliable, axiomatic, cut-and-reassemble engine for the homology groups. The next guide turns that engine on a CW complex to read off Betti numbers and the Euler characteristic, where the same exact-sequence logic becomes a finite, almost mechanical calculation.