Chain homotopy

Definition

Given two chain complexes $A$ and $B$, and chain maps $f,g:A\to B$, an algebraic homotopy or chain homotopy between $f$ and $g$ is an expression of $f-g$ as $dk+kd$ where $k$ is a collection of homomorphisms from $A_n$ to $B_{n+1}$ for every $n$.

Equivalently, two homomorphisms between chain complexes are in algebraic homotopy if they lie in the same coset of the group of homomorphisms of the form $dk+kd$.

If a chain homotopy exists between $f$ and $g$ we say that $f,g$ are chain-homotopic chain maps.

Facts

If $f$ and $g$ are two homotopic maps between topological spaces, then the induced maps between the singular complexes are in algebraic homotopy. For full proof, refer: Homotopy of maps induces chain homotopy