DeRhamCohomology 2004-06-12 18:01:09 2007-10-09 10:02:36 Definition de Rham cohomology group % 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 \newcommand{\HdR}{{\rm H}_{\rm dR}} \newcommand{\im}{\mathop{\mathrm{im}}} Let $X$ be a paracompact ${\cal C}^\infty$ differential manifold. Let $$ \Omega X=\bigoplus_{i=0}^\infty\Omega^i X $$ denote the graded-commutative $\mathbb{R}$-algebra of differential forms on $X$. Together with the exterior derivative $$ d^i\colon\Omega^i X\to\Omega^{i+1}X\quad(i=0,1,\ldots), $$ $\Omega X$ forms a chain complex $(\Omega X,d)$ of $\mathbb{R}$-vector spaces. The \PMlinkescapetext{{\it de Rham cohomology groups}} ${\rm H}_{\rm dR}^i X$ of $X$ are defined as the homology groups of this complex, that is to say $$ {\rm H}_{\rm dR}^i X:=(\ker d^i)/(\im d^{i-1})\quad(i=0,1,\ldots), $$ where $\Omega^{-1}X$ is taken to be 0, so $d^{-1}\colon 0\to\Omega^0 X$ is the zero map. The wedge product in $\Omega X$ induces the structure of a graded-commutative $\mathbb{R}$-algebra on $$ {\rm H}_{\rm dR}X:=\bigoplus_{i=0}^{\infty}\HdR^i X. $$ If $X$ and $Y$ are both paracompact ${\cal C}^\infty$ manifolds and $f\colon X\to Y$ is a differentiable map, there is an induced map $$ f^*\colon \HdR Y\to\HdR X, $$ defined by $$ f^*[\omega]:=[f^*\omega]\quad\hbox{for $\omega\in\ker d$}. $$ Here $[\omega]$ denotes the class of $\omega$ modulo $\im d$, and the second $f^*$ is the map $\Omega Y\to\Omega X$ induced by the functor $\Omega$. This action on differentiable maps makes the de Rham cohomology into a contravariant functor from the category of paracompact ${\cal C}^\infty$ manifolds to the category of graded-commutative $\mathbb{R}$-algebras. It turns out to be homotopy invariant; this implies that homotopy equivalent manifolds have isomorphic de Rham cohomology.