Relative Homology and Exactness

We present a proof of the long exact sequence in homology that is simpler than the usual proof, with no diagram chasing.

Relative homology Let A be any subspace of a space X, with inclusion i: A ⊂ X. We have the inclusion i_{#}: C_{∗}(A) ⊂ C_{∗}(X) of chain complexes. As usual, we write Z_{n}(X) for the group of n-cycles on X and B_{n}(X) for the group of n-boundaries.

The relative homology groups H_{n}(X, A) are defined as the homology groups of the quotient chain complex C_{∗}(X, A), where C_{n}(X, A) = C_{n}(X)/C_{n}(A), with n-cycles Z_{n}(X, A) and n-boundaries B_{n}(X, A). The inclusion j: (X, ∅) → (X, A) induces the quotient chain map j_{#}: C_{∗}(X) → C_{∗}(X, A). However, this is not always the most useful description of relative homology.

Definition 1 A relative n-cycle is an n-chain γ ∈ C_{n}(X) that satisfies ∂γ ∈ C_{n−1}(A). These form a subgroup of C_{n}(X) that we call Z′_{n}(X, A). We observe that

Z′_{n}(X, A) = (j_{#})^{−1}(Z_{n}(X, A)).  (2)

We write j′: Z_{n}(X) ⊂ Z′_{n}(X, A) for the obvious inclusion, which lifts j_{#}.

Similarly, we define the group B′_{n}(X, A) = (j_{#})^{−1}(B_{n}(X, A)) of relative nboundaries and note that

B′_{n}(X, A) = B_{n}(X) + C_{n}(A).  (3)

It follows that j_{#} induces the isomorphism

\frac{Z′_{n}(X, A)}{B′_{n}(X, A)} ≅ \frac{Z′_{n}(X, A)}{C_{n}(A)}/\frac{B′_{n}(X, A)}{C_{n}(A)} ≅ \frac{Z_{n}(X, A)}{B_{n}(X, A)} = H_{n}(X, A),  (4)

which allows us to use the groups Z′_{n}(X, A) and B′_{n}(X, A) to compute H_{n}(X, A).

Exactness We consider the commutative diagram

\xymatrix{C_{n}(A) \ar[r]^{i_{#}}_{⊂} & C_{n}(X) \ar[r]^{=} & C_{n}(X) & C_{n−1}(A) \ar[r]^{i_{#}}_{⊂} & C_{n−1}(X) \\ Z_{n}(A) \ar[u]_{∪} \ar[r]^{i_{#}}_{⊂} & Z_{n}(X) \ar[u]_{∪} \ar[r]^{j′}_{⊂} & Z′_{n}(X, A) \ar[u]_{∪} \ar[r]^{∂} & Z_{n−1}(A) \ar[u]_{∪} \ar[r]^{i_{#}}_{⊂} & Z_{n−1}(X) \ar[u]_{∪} \\ B_{n}(A) \ar[u]_{∪} \ar[r]^{i_{#}}_{⊂} & B_{n}(X) \ar[u]_{∪} \ar[r]^{j′}_{⊂} & B′_{n}(X, A) \ar[u]_{∪} \ar[r]^{∂} & B_{n−1}(A) \ar[u]_{∪} \ar[r]^{i_{#}}_{⊂} & B_{n−1}(X) \ar[u]_{∪}}  (5)

When we take the quotient of the second row by the third, we find the sequence

H_{n}(A) − ; − ; →^{i_{∗}} H_{n}(X) − ; − ; →^{j_{∗}} H_{n}(X, A) − ; − ; →^{∂} H_{n−1}(A) − ; − ; →^{i_{∗}} H_{n−1}(X)  (6)

of homology groups and homomorphisms, with the help of (4). Here, the connecting homomorphism ∂: H_{n}(X, A) → H_{n−1}(A) is canonical and not at all mysterious.

We make six observations about diagram (5); the first three are quite trivial.

1. If α ∈ Z_{n}(A), we have j′i_{#}α = α ∈ C_{n}(A) ⊂ B′_{n}(X, A).

2. If β ∈ Z_{n}(X), we have ∂j′β = ∂β = 0.

3. If γ ∈ Z′_{n}(X, A), we have i_{#}∂γ = ∂γ ∈ B_{n−1}(X).

4. Conversely, suppose β ∈ Z_{n}(X) satisfies j′β ∈ B′_{n}(X, A). By (3), β = ∂θ + α, where θ ∈ C_{n+1}(X) and α ∈ C_{n}(A). Further, α ∈ Z_{n}(A), since ∂α = ∂β − ∂∂θ = 0. We have i_{#}α = β − ∂θ, which is homologous to β. Thus in homology, i_{∗}[α] = [β].

5. If γ ∈ Z′_{n}(X, A) satisfies ∂γ ∈ B_{n−1}(A), i. e. ∂γ = ∂α for some α ∈ C_{n}(A), we have ∂(γ − α) = 0. So γ − α ∈ Z_{n}(X) differs from the given γ by α ∈ C_{n}(A) ⊂ B′_{n}(X, A).

6. Suppose α ∈ Z_{n−1}(A) satisfies i_{#}α ∈ B_{n−1}(X), i. e. α = ∂γ for some γ ∈ C_{n}(X). Then ∂γ = α shows that γ ∈ Z′_{n}(X, A) and lifts α.

These observations show that (6) is an exact sequence, i. e. Im i_{∗} = Ker j_{∗}, Im j_{∗} = Ker ∂, and Im ∂ = Ker i_{∗}. When we allow n to vary, we obtain the main result.

Theorem 7 For any pair of spaces (X, A), we have the homology long exact sequence of the pair (X, A),

… − ; − ; → H_{n}(A) − ; − ; →^{i_{∗}} H_{n}(X) − ; − ; →^{j_{∗}} H_{n}(X, A) − ; − ; →^{∂} H_{n−1}(A) − ; − ; →^{i_{∗}} H_{n−1}(X) − ; − ; →^{∂} …

General chain complexes The above proof used no special properties of the chain complex C_{∗}(X). Take any short exact sequence of chain complexes and chain maps

0 − ; − ; → A − ; − ; →^{i} B − ; − ; →^{j} C − ; − ; → 0,  (8)

so that each 0 → A_{n} → B_{n} → C_{n} → 0 is a short exact sequence. Then i identifies A with a sub-chain complex iA ⊂ B, and we may identify the quotient B/iA with C. We can therefore generalize Theorem 7, with no change in proof. (For example, we may as well replace A by iA ⊂ B and put Z′_{n}(C) = {β ∈ B_{n} : ∂β ∈ iA_{n−1}}.)

Theorem 9 Given the short exact sequence of chain complexes (8), we have the homology long exact sequence

… − ; − ; → H_{n}(A) − ; − ; →^{i_{∗}} H_{n}(B) − ; − ; →^{j_{∗}} H_{n}(C) − ; − ; →^{∂} H_{n−1}(A) − ; − ; →^{i_{∗}} H_{n−1}(B) − ; − ; → …