Example

Let $C$ be a complex. Consider the canonical exact sequence

(I)

and let $i : B(C) \to Z(C)$ be the canonical injection. Then the linking homomorphism $\partial(j, \delta) : \mathrm{H}(B(C)) (-1) \to \mathrm{H}(Z(C)) (-1)$ identifies with $\mathrm{H}(i) (-1)$, as one verifies at once. Since $\mathrm{H}(\delta) = 0$, the homology exact sequence associated to (I) decomposes into the short exact sequences

(II_n)

\* Mappings :

1) Singular homology

Let $A$ be a ring. For any topological space $X$, we define the singular complex $C(X, A)$ of $X$ with coefficients in $A$ as follows:

In $\mathbf{R}^{(N)}$, denote by $(e_n)$ the canonical basis; we call the $n$-canonical simplex the convex hull $\Delta_n$ of $\{ e_0, ..., e_n \}$. For $i \in \{ 0, ..., n \}$, we define the affine mapping $\iota_i : \Delta_{n-1} \to \Delta_n$ by $\iota_i(e_k) = e_k$ for $k < i$ and $\iota_i(e_k) = e_{k+1}$ for $k \geq i$. We denote $C_n(X, A)$ the a-module $A^{(\Sigma_n(X))}$, where $\Sigma_n(X)$ is the set of continuous mappings from $\Delta_n$ into $X$; for $n < 0$, we set $C_n = 0$. For $i \in \{ 0, ..., n \}$, we define the linear mapping $\partial_{n, i} : C_n(X, A) \to C_{n-1}(X, A)$ by $\partial_{n, i}(e_s) = e_{s \circ \iota_i}$ for $s \in \Sigma_n(X)$, and we set $d_n = \Sigma (-1)^i \partial_{n, i}$. We verify that

is a complex. Its homology is called the singular homology of $X$ with coefficients in $A$ and is denoted $\mathrm{H}(X, A)$ or simply $\mathrm{H}(X)$.

If $Y$ is a subspace of $X$, we denote $C(X, Y, A)$ the quotient complex $C(X, A)/C(Y, A)$, and $\mathrm{H}(X, Y, A)$ its homology. It follows from Theorem 1 that there is an exact sequence:

2) Finite cell complexes

Let $X$ be a separated topological space. A finite cell decomposition of $X$ is given by an increasing sequence $(X_n)_{n \in \mathbf{Z}}$ of closed subspaces of $X$ satisfying the following conditions: (i) $X_n = \varnothing$ for $n < 0$; (ii) there exists an $N$ such that $X_N = X$ (hence $X_n = X$ for $n > N$); (iii) for every $n$, the space $X_n - X_{n-1}$ has only a finite number of connected components, called the $n$-dimensional cells; (iv) for every $n$, and for every connected component $C$ of $X_n - X_{n-1}$, there exists a homeomorphism from the open $n$-dimensional euclidean ball $\hat{B}_n$ (TG, VI, p. 10) onto $C$ that extends to a continuous mapping of the closed ball into $X$.

One can show that these conditions imply that $\mathrm{H}_n(X_n, X_{n-1}, A)$ is a free a-module $\Gamma_n$ of rank equal to the number of $n$-dimensional cells, and that $\mathrm{H}_i(X_n, X_{n-1}, A) = 0$ for $i \neq n$. We have $C(X_n, X_{n-1}, A) = C(X_n, X_{n-2}, A)/C(X_{n-1}, X_{n-2}, A)$, hence an exact sequence

involving a linking homomorphism $d_n : \Gamma_n \to \Gamma_{n-1}$. We have $d_n \circ d_{n+1} = 0$, which allows us to define a complex $\Gamma : \cdots \to \Gamma_n \xrightarrow{d_n} \Gamma_{n-1} \to \cdots$.

The exact sequence

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(Note: The final line "The exact sequence" is left as part of the text since the source ends there.)

shows by induction on p that $H_n(X_p) = 0$ for $p < n$, that $H_n(X_n) = \mathrm{Ker}\,(d_n : \Gamma_n \to \Gamma_{n-1})$ and that $H_n(X_p) = H_n(\Gamma)$ for $p > n$. In particular $H_n(X)$ is identified with $H_n(\Gamma)$.