chain homotopy

Last updated January 27, 2022

Definition

Let $f,g:C_\bullet\to D_\bullet$ be maps between chain complexes.

$f$ and $g$ are homotopic, denoted $f\simeq g$, if there exist maps $h_\bullet$ such that

The idea is that $f$ and $g$ induce the same map on homology. $f_n$ and $g_n$ differ only by a boundary: given a cycle $z\in C_n$, $(f_n-g_n)(z)\in\text{Im }d_{n+1}$.