Definition of chain homotopic.

Question

Let $C_{\bullet}, D_{\bullet}$ be two nonnegatively graded chain complexs of $R$-modules with maps $d^C,d^D$ respectively($d^C_n: C_{n+1} \to C_n$), and let $f,g: C_{\bullet} \to D_{\bullet}$ be two chain maps. Then, a chain homotopy from $f$ to $g$ is a sequence of $R$-linear maps $h_n: C_{n-1} \to D_{n}$ such that $$f_n - g_n = d^D_{n}h_{n+1} + h_{n}d^{C}_{n-1}.$$

First of all, is my definition correct? My second question is, what does this really mean? My notes say that if such a chain homotopy exist then we say that $f$ and $g$ are chain homotopic and we write $f \cong g$. I feel that the definition is some kind of "isomorphism" between two chain maps but I cant see how, can someone explain what it really means to have a chain homotopy, i cant really "unfold" the definition.

Answer 1

I would post this as a comment but unfortunately I don't have enough reputation at the moment. Yes, your definition is correct, as is your appearent intuition. Chain homotopies are a concept which originates from topology, where the complexes at hand are usually motivated by a decomposition of a topological space into cells (CW-complexes), simplices (simplicial complexes) or continuous deformations of simplicies (singular homology) hence the nomenclature. And what you suspect is true: chain homotopies are somewhat related to isomorphisms of chain complexes. Given two maps {$f_n$}$_n$ and {$g_n$}$_n$ between chain complexes $C_*$ and $D_*$ which are chain homotopic (i.e. there exists a chain homotopy between them), then they induce the same map after passing to homology. Their importance comes from the idea that what you would expect an isomorphism of chain complexes to be (namely a level-wise isomorphism which commutes with differentials) is a very restrictive definition. Instead, what one considers is quasi-isomorphisms, which are morphisms of chain complexes which induce isomorphisms in homology. And this is precisely where chain homotopies come in handy as not all quasi-isomorphisms must necessarily be given by isomorphisms of chain complexes.