Chain homotopy

Remarks

Chain homotopy is an equivalence relation compatible with composition. A chain map $f$ is a chain homotopy equivalence if there is a chain map $g \colon D_\bullet \to C_\bullet$ with $g \circ f \simeq 1_C$ and $f \circ g \simeq 1_D$. A complex is contractible if its identity is chain homotopic to the zero map. In an abelian category, chain homotopic maps induce the same morphisms on homology: a cycle $z$ is sent by $f - g$ to $d^D h(z)$, which is a boundary.

Example: let $C_\bullet = D_\bullet$ have $C_1 = C_0 = \Z$, $d_1 = 1_{\Z}$, and $C_n = 0$ otherwise; let $f$ be the identity and $g = 0$. Take $h_0 = 1_{\Z} \colon C_0 \to D_1$ and all other $h_n = 0$. In degree $0$, $d_1 h_0 + h_{-1} d_0 = 1 + 0 = 1 = f_0 - g_0$; in degree $1$, $d_2 h_1 + h_0 d_1 = 0 + 1 = 1 = f_1 - g_1$.

Non-example: let $C_\bullet = D_\bullet$ be $\Z$ in degree $0$ and $0$ elsewhere. In degree $0$, the map $h_0$ lands in $D_1 = 0$ and $d^C_0$ lands in $C_{-1} = 0$, so $d_1 h_0 + h_{-1} d_0 = 0$, while $f_0 - g_0 = 1$. So the identity is not chain homotopic to $0$; indeed it induces the identity of $H_0 = \Z$.