Chain homotopy

Definition

Let $f,g:C_\bullet\to D_\bullet$ be chain maps between chain complexes $(C_\bullet,d^C)$ and $(D_\bullet,d^D)$. A chain homotopy from $f$ to $g$ is a family of $R$-linear maps

such that, for every $n$,

One writes $f\simeq g$ if such an $s$ exists.

Key consequence

If $f\simeq g$, then the induced maps on homology agree:

where homology module is used.

Cross-links

• Special case: a contracting homotopy $\mathrm{id}\simeq 0$ shows a complex is “contractible,” hence exact .
• Chain homotopy is the basic equivalence relation behind chain-homotopy categories; compare chain map .

Examples

1. Degree-0 complexes: homotopy forces equality.
   If $C_\bullet$ and $D_\bullet$ are concentrated in degree $0$, then any $s_n$ must be $0$ (there is no $D_{1}$), so the homotopy identity becomes $f_0-g_0=0$. Thus $f\simeq g$ implies $f=g$ in this case.

2. A contractible 2-term complex.
   Consider the complex $C_\bullet$ with $C_1=R$, $C_0=R$, and $d_1=\mathrm{id}_R$ (all other $C_n=0$). This is a chain complex since $d_0=0$.
   Define $s_0:C_0\to C_1$ to be $\mathrm{id}_R$ and all other $s_n=0$. Then for $n=0$,

   $$(\mathrm{id}-0)_{0} = d_1 s_0 + s_{-1} d_0 = \mathrm{id}\circ \mathrm{id} + 0,$$

   and for $n=1$,

   $$(\mathrm{id}-0)_{1} = d_2 s_1 + s_0 d_1 = 0 + \mathrm{id}\circ \mathrm{id}.$$

   Hence $\mathrm{id}_{C_\bullet}\simeq 0$. In particular $H_n(C_\bullet)=0$ for all $n$, so $C_\bullet$ is exact .

3. Split exact complexes admit contracting homotopies.
   If each short exact sequence

   $$0\to \operatorname{im}(d_{n+1}) \to C_n \to \operatorname{im}(d_n)\to 0$$

   splits (as in many algebraic settings), then one can choose splittings to build maps $s_n:C_n\to C_{n+1}$ with $\mathrm{id}\simeq 0$. This provides a conceptual way to recognize contractible (hence exact) complexes.