chain homotopy

Definition

Given chain maps $f_{\bullet },g_{\bullet }:A_{\bullet }\to B_{\bullet }$ between chain complexes, a chain homotopy is a sequence of group homomorphims

$h_n:A_n⟶B_{n+1}$

such that

$f_n-g_n=d\circ h_n+h_{n-1}\circ d$

Seen a little clearer in the diagram below

Connection to homology

Chain homotopic maps induce the same homomorphims of homology groups.

Proof

todo