Definition

Given chain maps between chain complexes, a chain homotopy is a sequence of group homomorphims

such that

Seen a little clearer in the diagram below

Connection to homology

Chain homotopic maps induce the same maps on homology.

It suffices to prove that

as maps .

Let with representative (this means that ).

Then

The homotopy condition gives

Let . Consider the complex

(note this a -linear resolution of ).

we have a chain map given by multiplication by in each degree. is null-homotopic (homotopic to the 0 map).

Thus we have the following diagram for the homotopy