Example

Let $u : M \to N$ be a homomorphism of A-modules; then the only non-zero homogeneous components of $\mathrm{Cyl}\,(u)$ and $\mathrm{Con}\,(u)$ are

and we have $\overline{D}(m) = (m, -u(m))$, $D(m) = -u(m)$ for $m \in M$; consequently,

Let $X$ and $Y$ be two topological spaces and $f$ a continuous mapping from $X$ into $Y$. The cylinder of $f$ is called the quotient space $\mathrm{Cyl}\,(f)$ of the topological sum of $[0, 1] \times X$ and $Y$ by the equivalence relation identifying the point $(1, x)$ of $[0, 1] \times X$ with the point $f(x)$ of $Y$ for every $x \in X$. The cone of $f$ is called the quotient space $\mathrm{Con}\,(f)$ of the topological sum of a space reduced to a point $s$ and $\mathrm{Cyl}\,(f)$ by the equivalence relation identifying $s$ with the image of $(0, x)$ for every $x \in X$: we still denote by $s$ the image of $s$ in $\mathrm{Con}\,(f)$.

Suppose that $X$ and $Y$ are endowed with cellular decompositions $(X_n)$ and $(Y_n)$ (cf. No. 3), and suppose that $f(X_n) \subset Y_n$ for every $n$. We obtain a cellular decomposition $(S_n)$ of $\mathrm{Cyl}\,(f)$ (resp. of $\mathrm{Con}\,(f)$) by taking for $S_n$ the image of $(\{0\} \times X_n) \cup ([0, 1] \times X_{n-1}) \cup Y_n$

Let $\Gamma(X), \Gamma(Y), \Gamma(\mathrm{Cyl}\,(f)), \Gamma(\mathrm{Con}\,(f))$ denote the complexes associated with these cellular decompositions.

The complex $\Gamma(s)$ associated with the space $\{ s \}$ endowed with its unique cellular decomposition is reduced to the module $\mathbf{A}$ and is identified with a subcomplex of $\Gamma(\mathrm{Con}\,(f))$; let $\Gamma(\mathrm{Con}\,(f),\,s)$ denote the quotient complex. The mapping $f$ defines a morphism of complexes $\Gamma(f) : \Gamma(X) \to \Gamma(Y)$, and one can show that the complexes $\Gamma(\mathrm{Cyl}\,(f))$ and $\Gamma(\mathrm{Con}\,(f),\,s)$ are identified respectively with $\mathrm{Cyl}\,(\Gamma(f))$ and $\mathrm{Con}\,(\Gamma(f))$.

Par ailleurs, et sans hypothèses sur $X$ et $Y$, on associe à $f$ un morphisme de complexes $f_* : C(X,\,\mathbf{A}) \to C(Y,\,\mathbf{A})$. On peut construire des homotopismes injectifs de $\mathrm{Cyl}\,(f_*)$ dans $C(\mathrm{Cyl}\,(f),\,\mathbf{A})$ et de $\mathrm{Con}\,(f_*)$ dans $C(\mathrm{Con}\,(f),\,\{s\},\,\mathbf{A})$. Soient $\tilde{f} : X \to \mathrm{Cyl}\,(f)$ l'application qui à $x$ associe l'image de $(0,\,x)$, $\alpha : Y \to \mathrm{Cyl}\,(f)$ l'application canonique et $\beta : \mathrm{Cyl}\,(f) \to Y$ l'application qui associe $y$ à son image dans $\mathrm{Cyl}\,(f)$ pour $y \in Y$ et $f(x)$ à l'image de $(t,\,x)$ dans $\mathrm{Cyl}\,(f)$ pour $t \in [0,\,1]$ et $x \in X$. L'application $\tilde{f}$ est un homéomorphisme de $X$ sur un fermé de $\mathrm{Cyl}\,(f)$, on a $\beta \circ \alpha = \mathrm{Id}_Y$, et $\alpha \circ \beta$ est topologiquement homotope à l'identité de $\mathrm{Cyl}\,(f)$. Ces propriétés sont à rapprocher de la proposition 7 ci-dessous. \*

Considérons maintenant les applications $\mathbf{A}$-linéaires graduées de degré 0