ChainHomotopy 2002-01-23 13:04:24 2006-06-06 02:06:19 Definition chain homotopic % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics \usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let $(A,d)$ and $(A^{'},d^{'})$ be chain complexes and $f:A \to A^{'}$, $g:A \to A^{'}$ be chain maps. A \emph{chain homotopy} $D$ between $f$ and $g$ is a sequence of homomorphisms $\{D_{n}:A_{n} \to A_{n+1}^{'}\}$ so that $d_{n+1}^{'} \circ D_{n} + D_{n-1} \circ d_{n}=f_{n}-g_{n}$ for each $n$. Thus, we have the following diagram: $$ \xymatrix{ & A_{n+1} \ar[r]^{d_{n+1}} \ar[d]_{f_{n+1}-g_{n+1}} & A_{n} \ar[dl]^{D_{n}} \ar[r]^{d_{n}} \ar[d] & A_{n-1} \ar[dl]_{D_{n-1}} \ar[d]^{f_{n-1}-g_{n-1}} \\ & A_{n+1}^{'} \ar[r]_{d_{n+1}^{'}} & A_{n}^{'} \ar[r]_{d_{n}^{'}} & A_{n-1}^{'} } $$ If there exists a chain homotopy between $f$ and $g$, then $f$ and $g$ are said to be \emph{chain homotopic.}