Chain homotopy

Definition

Given two chain complexes $A$ and $B$, and homomorphisms $f,g:A\to B$, an algebraic homotopy between $f$ and $g$ is an expression of $f-g$ as $dk-kd$ where $k$ is some homomorphism from the complex $A$ to the complex $B$.

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 $dk-kd$.

Facts

if $f$ and $g$ are two homotopic maps between topological spaces, then the induced maps between the singular complexes are in algebraic homotopy. Fill this in later