If two continuous maps $f, g : X \to Y$ are homotopic, then the induced chain maps $f_\sharp, g_\sharp : C_\bullet(X) \to C_\bullet(Y)$ are chain homotopic. This is the fundamental connection between topological and algebraic homotopy. The chain homotopy is constructed from the prism operator.

A complex $A^\bullet$ is contractible if $\mathrm{id}_{A^\bullet} \sim 0$, equivalently if $A^\bullet$ is homotopy equivalent to the zero complex. A contractible complex is acyclic ($H^n = 0$ for all $n$), but not every acyclic complex is contractible.

Any two projective resolutions of a module $M$ are chain homotopy equivalent. This is the comparison theorem: if $P_\bullet \to M$ and $Q_\bullet \to M$ are projective resolutions, there exist chain maps $f : P_\bullet \to Q_\bullet$ and $g : Q_\bullet \to P_\bullet$ (lifting $\mathrm{id}_M$) with $g \circ f \sim \mathrm{id}_P$ and $f \circ g \sim \mathrm{id}_Q$.

Since projective resolutions are unique up to chain homotopy, and homotopic maps induce equal maps on cohomology, the derived functors $R^nF(M) = H^n(F(I^\bullet))$ (using an injective resolution $M \to I^\bullet$) are well-defined up to canonical isomorphism.

The bar resolution $B_\bullet \to \mathbb{Z}$ for a group $G$ admits an explicit contracting homotopy $s_n : B_n \to B_{n+1}$ defined by $s_n[g_1|\cdots|g_n] = [1|g_1|\cdots|g_n]$. This shows $B_\bullet$ is a resolution (acyclic augmented complex).

A map $f : P_\bullet \to Q_\bullet$ between projective resolutions is null-homotopic iff it induces the zero map on $\mathrm{Ext}$. More precisely, elements of $\mathrm{Ext}^n(M, N)$ are chain homotopy classes of chain maps $P_\bullet \to I_\bullet[n]$.

$f \sim g$ if and only if $f - g$ is null-homotopic, which happens iff the mapping cone $\mathrm{Cone}(f - g)$ is contractible. This connects chain homotopy to the cone construction.

The Poincare lemma states that on a contractible open set $U \subseteq \mathbb{R}^n$, the de Rham complex is acyclic. The proof constructs an explicit chain homotopy using integration along fibers.

In the 2-category of complexes, chain homotopies are the 2-morphisms between chain maps (1-morphisms). This perspective leads to the DG category structure on complexes and ultimately to stable infinity-categories.

In $\mathbb{Z}\text{-}\mathbf{Mod}$, the complex $0 \to \mathbb{Z} \xrightarrow{2} \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0$ is acyclic (it is a short exact sequence) but not split, hence not contractible. The obstruction to contractibility lives in $\mathrm{Ext}^1$.

Chain homotopy $\sim$ is an equivalence relation on $\mathrm{Hom}_{\mathbf{Ch}}(A^\bullet, B^\bullet)$ compatible with composition: if $f \sim g$ and $f' \sim g'$, then $f' \circ f \sim g' \circ g$. The null-homotopic maps form a two-sided ideal.

In a DG category, homotopy between morphisms is formalized by the condition $f - g = d(s)$ in the DG hom-complex. This generalizes chain homotopy and is the foundation for DG enhancements of triangulated categories.