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 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.